Previous chemotherapy with at least 3 cycles of docetaxel for hormone-sensitive metastatic prostate cancer Prior treatment with docetaxel within 6 months of study enrollment taxanes (paclitaxel or docetaxel) or epirubicin, Docetaxel monotherapy is a reasonable treatment in the judgement of the Investigator Prior use of abiraterone acetate, enzalutamide, docetaxel, cabazitaxel, carboplatin, or radium-223 for the treatment of castration-resistant disease\r\n* Prior docetaxel use in the hormone-sensitive disease setting is allowed, but must be completed >= 4 weeks prior to enrollment\r\n* Prior sipuleucel-T use is allowed, but must be completed >= 4 weeks prior to enrollment\r\n* Concurrent use of zolendronic acid or denosumab is allowed on study Prior systemic chemotherapy (docetaxel, cabazitaxel, estramustine, other cytotoxic agents) Prior treatment with docetaxel chemotherapy in the castration-resistant setting. Prior treatment with docetaxel in either the neoadjuvant or adjuvant setting or for hormone sensitive disease (e.g., CHAARTED population) is allowed, as long as therapy was completed > 12 months prior to study registration Agree to use barrier methods of birth control during the docetaxel portion of the protocol and for at least one month after last docetaxel administration Any previous exposure to docetaxel Prior docetaxel for hormone-sensitive prostate cancer is permitted if =< 6 doses were given in conjunction with first-line androgen deprivation therapy and > 12 months since last dose of docetaxel Prior treatment with docetaxel or cabazitaxel for metastatic castration-resistant prostate cancer is prohibited Patients planned to receive docetaxel with contra-indications to receive docetaxel Previous treatment with docetaxel for metastatic prostate cancer Has received any prostate cancer directed chemotherapy in the castration resistant setting. Subjects who have received up to 6 prior doses of docetaxel in the castration sensitive setting are permitted if they have not experienced disease progression within 36 weeks of last treatment with docetaxel Has known grade >= 3 docetaxel-related toxicities or docetaxel toxicity related dose interruption or discontinuation Patients who have had chemotherapy for metastatic castration-resistant prostate cancer within the past year (patients who have had docetaxel for metastatic castration sensitive per CHAARTED data may enroll as long as they did not have progressive disease while on docetaxel and are 6 months removed from treatment, with all treatment related toxicities resolving to at least grade 1) Chemotherapy naive (prior docetaxel chemotherapy [as per chemohormonal therapy versus androgen ablation randomized trial for extensive disease (CHAARTED) data] for castration sensitive disease is allowed) Prior chemotherapy (e.g. docetaxel, cabazitaxel) for CRPC; prior docetaxel administered in the castrate-sensitive space is allowed Prior treatment with ADI-PEG 20, gemcitabine, or docetaxel Prior systemic chemotherapy (docetaxel, cabazitaxel, estramustine, other cytotoxic agents) Prior taxane-based chemotherapy with progressive disease on chemotherapy\r\n* Prior docetaxel for metastatic hormone sensitive prostate cancer is allowed, if no progression of disease on docetaxel as defined by RECIST v1.1 and PCWG3\r\n* Prior taxane-based chemotherapy (i.e. docetaxel or cabazitaxel with or without platinum agent) for mCRPC is allowed if no progression of disease on chemotherapy as defined by RECIST v1.1 and PCWG3 Docetaxel appropriate\r\n* Patients who have not received prior docetaxel (or other taxane therapy) in the advanced setting are eligible for all cohorts\r\n* Patients who have previously received docetaxel (or other taxane therapy) in the advanced setting are eligible for the dose escalation cohort only, if anticipated to have maintained taxane sensitivity and in the opinion of the investigator would still benefit from further docetaxel therapy Has had any previous exposure to paclitaxel or docetaxel in the last 5 years. Radiographic progressive disease, irrespective of PSA changes, after receiving at least 4 cycles of docetaxel or cabazitaxel Docetaxel: Creatinine clearance no minimum Received more than 10 cycles of docetaxel (for docetaxel cohort only) or 6 of cabazitaxel (for cabazitaxel cohort only) Last docetaxel or cabazitaxel dose > 6 weeks prior to enrollment Progressive disease, both docetaxel naive and docetaxel treated. Less than 6 weeks since last myelosuppressive therapy (including Docetaxel, Cabazitaxel, 223Ra, 153Sm) or other radionuclide therapy. Prior treatment with chemotherapy (docetaxel or cabazitaxel) for castration resistant prostate cancer A minimum of 4 weeks from prior chemotherapy, including but not limited to, docetaxel, cabazitaxel, mitoxantrone, carboplatin, cisplatin, or estramustine; if applicable, prior to registration Previous treatment with docetaxel or an Axl inhibitor Patients who had received at least 2 prior therapies for advanced or metastatic disease condition, including platinum doublet and pemetrexed, docetaxel, or immunotherapy, and were refractory to or unable to tolerate their last prior therapy mCRPC EXPANSION COHORT: Patients must have received prior treatment with enzalutamide and/or abiraterone with the exception of patients who were treated with docetaxel and androgen deprivation therapy for metastatic castrate-sensitive prostate cancer and progressed on docetaxel treatment or who progress within one month of the last docetaxel dose mCRPC EXPANSION COHORT: Patients who were treated for metastatic castrate-sensitive prostate cancer with docetaxel and androgen deprivation therapy who progress on docetaxel treatment or who progress within one month of the last docetaxel dose are eligible Received any prior treatment with any taxane (docetaxel or paclitaxel) for small cell lung cancer Subjects must be considered suitable for chemotherapy with either single-agent pemetrexed or docetaxel. NSCLC that is predominantly squamous cell carcinoma, and patient had docetaxel as part of his prior chemotherapy. Prior therapy with abiraterone, enzalutamide and/or docetaxel; if a patient has not received docetaxel or cabazitaxel chemotherapy, the patient must be informed of this treatment choice as an alternative; if the patient has received docetaxel or cabazitaxel chemotherapy or refuses one of both of these therapies, this rationale must be documented and the patient is then eligible; patient must be offered and made aware of all Food and Drug Administration (FDA)?approved treatment options; patients with bone only disease may not have received radium-223 Prior cytotoxic chemotherapy (for example, but not limited to, docetaxel, mitoxantrone, cabazitaxel) within 6 months of registration is prohibited ARM B COHORT 2: Patients must not have prior exposure to docetaxel ARM B COHORT 3: Patients must not have prior exposure to docetaxel PHASE II STUDY METASTATIC CASTRATE-RESISTANT PROSTATE CANCER COHORT 4 ELIGIBILITY CRITERIA (MEDI+O ONLY):\r\nPatients must have received prior treatment with enzalutamide and/or abiraterone with the exception of patients who were treated with docetaxel and androgen deprivation therapy for metastatic castrate-sensitive prostate cancer and progressed on docetaxel treatment or who progress within one month of the last docetaxel dose PHASE II STUDY METASTATIC CASTRATE-RESISTANT PROSTATE CANCER COHORT 4 ELIGIBILITY CRITERIA (MEDI+O ONLY):\r\nPatients who were treated for metastatic castrate-sensitive prostate cancer with docetaxel and androgen deprivation therapy who progress on docetaxel treatment or who progress within one month of the last docetaxel dose are eligible Prior treatments with Cyp 17 inhibitors like TAK-700/orteronel, ketoconazole, radium 223 or docetaxel (up to 6 cycles of docetaxel given in the non CRPC setting is allowed); prior treatment with sipuleucel-T is allowed To commence third line docetaxel, patients must have grade 2 or less neuropathy from prior oxaliplatin treatment; also bilirubin > upper limit of normal (ULN), or AST and/or ALT > 1.5 x ULN concomitant with alkaline phosphatase > 2.5 x ULN are not eligible for docetaxel therapy Prior docetaxel for hormone-sensitive prostate cancer is permitted if =< 6 doses were given in conjunction with first-line androgen deprivation therapy and > 12 months since last dose of docetaxel Prior treatment with docetaxel or cabazitaxel for metastatic castration-resistant prostate cancer is prohibited Patients who have had prior therapy with gemcitabine or docetaxel Prior therapy with docetaxel No prior chemotherapy for the treatment of mCRPC; patients may have received docetaxel for the treatment of hormone-sensitive prostate cancer Prior therapy with a MEK-inhibitor or docetaxel for metastatic non-small cell lung cancer (docetaxel in the adjuvant setting will be allowed) No prior docetaxel or cabazitaxel chemotherapy for metastatic castration-resistant prostate cancer (mCRPC) (men treated with prior docetaxel administered as up-front therapy with androgen deprivation therapy [ADT] > 6 months ago will be eligible); prior abiraterone is allowed Prior treatment with docetaxel or cabazitaxel for mCRPC Prior treatment with docetaxel. Patients must have progressed on Arm 2 (docetaxel) of this sub-study Patients must have progressed on Arm 2 (docetaxel) of this sub-study Patients must have progressed on Arm 2 (docetaxel) of this sub-study Prior treatment with docetaxel is allowed but not required Patients who are deemed to have high-risk or extensive metastatic, hormone sensitive prostate cancer (mHSPC) per “clinical judgment” of the treating physician are eligible for enrollment if they are unsuitable candidates for docetaxel or if they have declined docetaxel therapy Prior chemotherapy (for example, docetaxel, cabazitaxel) for treatment of CRPC, except when docetaxel has been given for hormone-sensitive prostate cancer For Cohort B: Has received prior treatment with docetaxel or another chemotherapy agent for metastatic prostate cancer For Cohort C: Has received prior chemotherapy for mCRPC. Prior docetaxel for metastatic hormone-sensitive prostate cancer is allowed if ?4 weeks elapsed from last dose of docetaxel Up to 6 cycles of docetaxel therapy with final treatment administration completed within 2 months of day 1 and no evidence of disease progression during or after the completion of docetaxel therapy; Participants who received docetaxel treatment must meet the following criteria: a) Received a maximum of 6 cycles of docetaxel therapy for mHSPC; b) Received the last dose of docetaxel <=2 months prior to randomization; c) Maintained a response to docetaxel of stable disease or better, by investigator assessment of imaging and PSA, prior to randomization Have not received prior treatment with docetaxel. Patients who have disease progression during, or after, receiving docetaxel and have had at least 12 weeks of treatment and in the opinion of the investigator are unlikely to derive significant benefit from additional docetaxel-based therapy, or were intolerant to therapy with this agent. Prior therapy with docetaxel for NSCLC Prior treatment with any of the chemotherapy medications (cisplatin or docetaxel) for HNSCC or with AZD1775 Prior cytotoxic chemotherapy for prostate cancer, but up to 2 cycles of docetaxel chemotherapy for metastatic disease is permitted. Previously received docetaxel or are not healthy enough per clinical judgment or declined to receive it Prior cytotoxic chemotherapy with the exception of docetaxel or cabazitaxel; treatment with docetaxel or cabazitaxel must be discontinued >= 4 weeks from the time of enrollment, and recovery of adverse events (AEs) to grade 1 or baseline (however, ongoing neuropathy is permitted) Patients must have one of the following a) low volume disease (defined as no visceral metastases and < 4 bone metastases) or b) are not candidates for docetaxel based chemotherapy or c) refused docetaxel chemotherapy For the randomised phase only, patients must have received chemotherapy in the form of docetaxel treatment for metastatic castration-resistant prostate cancer. Note: patients who discontinued docetaxel for toxicity reasons and without completing the full course will still be eligible to enter this study provided they received at least 2 cycles of chemotherapy. Prior disease progression on docetaxel or paclitaxel in metastatic setting Participants may have received prior docetaxel-based chemotherapy for prostate cancer; such chemotherapy must have been stopped at least three weeks prior to the first dosing in this study Patients who have had chemotherapy for metastatic castration-resistant prostate cancer; (patients who have had docetaxel for metastatic castration sensitive disease per ChemoHormonal Therapy Versus Androgen Ablation Randomized Trial for Extensive Disease in Prostate Cancer [CHAARTED] Data may enroll as long as they did not have progressive disease while on docetaxel and are 6 months removed from treatment, will all treatment related toxicities resolving to at least grade 1) Prior cytotoxic chemotherapy (e.g. docetaxel, mitoxantrone, cabazitaxel) within 6 months of registration is prohibited The subject has had treatment with docetaxel for the treatment of metastatic castrate-sensitive prostate cancer within 6 months before the first dose of study treatment The subject has had progression of prostate cancer during 6 cycles of prior docetaxel treatment for castrate sensitive disease Prior treatment with Docetaxel Prior therapy with pazopanib, gemcitabine or docetaxel; patients who have had prior treatment with gemcitabine or docetaxel for a prior malignancy are eligible if they meet the criteria in exclusion #3 and did not experience significant drug related toxicity Planned or recent initiation of standard docetaxel therapy; patients may be enrolled after receiving standard docetaxel therapy as long as the patient has not demonstrated evidence of progression for more than 45 days before enrollment (“late enrollers”) Known hypersensitivity to docetaxel, fluorouracil (5-FU) Patients who have received previous docetaxel or cisplatin Prior treatment with docetaxel-based chemotherapy Prior treatment with docetaxel within 6 months of enrollment Previous treatment with docetaxel. Hypersensitivity to the active substance or ingredients of PEGPH20 and docetaxel. Received prior treatment with docetaxel. Patients receiving chemotherapy (e.g., docetaxel, cabazitaxel, taxane, or platinum as single agents or in combination) as their cancer treatment Patient must be eligible for chemotherapy with docetaxel Symptomatic patients who, in the opinion of the investigator, may benefit from docetaxel-based chemotherapy Prior treatment with docetaxel or mogamulizumab; Prior treatment with docetaxel Prior docetaxel or other chemotherapy for mCRPC; patients who have received docetaxel for metastatic hormone-sensitive prostate cancer are eligible Prior treatment with MLN4924; however, prior treatment with docetaxel, paclitaxel, and carboplatin is allowed. Prior treatment with docetaxel for recurrent or advanced NSCLC Prior treatment with docetaxel-containing therapy Participants with NSCLC to be treated with docetaxel need to have received at least one prior anti-cancer treatment regimen in an advanced setting and to have docetaxel be considered appropriate treatment Received prior docetaxel chemotherapy Part 2 only: Patients must be docetaxel-naive. Part 2 only: Patients may not have had prior treatment with docetaxel. History of treatment with docetaxel in any setting. Participants treated with prior paclitaxel are eligible. Prior treatment with docetaxel-based chemotherapy Patients who have received docetaxel plus anti-androgen therapy (ADT) for metastatic castrate sensitive prostate cancer are eligible for the study; (patients may enroll as long as they did not have progressive disease while on docetaxel and are 6 months removed from treatment, with all treatment related toxicities resolving to at least grade 1) Have had treatment with docetaxel for the treatment of metastatic castrate-sensitive prostate cancer within 6 months before the first dose of study enrollment Have had progression of prostate cancer on prior docetaxel treatment for castrate sensitive disease Patients with prior docetaxel chemotherapy Pain appeared during or up to 12 weeks after treatment with oxaliplatin, paclitaxel, docetaxel or any combination of these Have metastatic castration-resistant prostate cancer (prostate cancer progressing despite castrate levels of testosterone [< 50 ng/dL] using standard measures of progression defined by Prostate Cancer Working Group 2), are chemo-naive for metastatic castration-resistant prostate cancer (mCRPC); patients must have symptomatic disease or visceral metastases or otherwise be eligible for docetaxel treatment per investigator judgment (e.g. for progression on imaging or rapidly rising PSA despite 2nd line hormonal treatment);\r\n* Note: Six cycles of prior docetaxel are allowed in hormone-sensitive disease, per Eastern Cooperative Oncology Group (ECOG) 3805 data and have been off of docetaxel for at least 12 months Undergoing current therapy for organ confined or systemic disease; this does not preclude patients who had previously received upfront docetaxel in the hormone sensitive setting Prior docetaxel-based chemotherapy is permitted but not required