Inclusion Criteria: Each patient must meet all of the following inclusion criteria to be enrolled in the study: - Female patients 18 years or older - Previously treated, metastatic or locally recurrent malignancy with 1 of the following diagnoses, which has been confirmed histologically or cytologically: adenocarcinoma of the breast (Phase 1 only), recurrent epithelial ovarian, fallopian tube, or primary peritoneal carcinoma (Phase 1 and 2) - In the Phase 1 portion of the study, patients with breast cancer must have received treatment with at least 1 but no more than 4 prior chemotherapy regimens not including regimens received in the neoadjuvant and/or adjuvant setting - Patients with breast cancer must have measurable disease per Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 - No antineoplastic therapy or radiotherapy within 3 weeks before enrollment (2 weeks for regimens with recovery expected within 7 to 14 days) and recovered from toxicities of prior therapy (except alopecia); the patient must have recovered from all treatment-related toxicities and must have evidence of PD or persistent disease - Eastern Cooperative Oncology Group (ECOG) performance status 0 or 1 - Adequate bone marrow, liver and renal function - Postmenopausal at least 1 year, OR Surgically sterile, OR If childbearing potential, agree to 2 effective methods of nonhormonal contraception, or agree to completely abstain from heterosexual intercourse - Able to provide written informed consent - Willing to comply with scheduled visits, treatment plan, laboratory tests and other trial procedures - Suitable venous access Specific Inclusion Criteria for Patients with Recurrent Ovarian, Fallopian Tube or Peritoneal Cancer: - Prior treatments must have included a platinum and a taxane; the most recent treatment need not be a platinum-containing or taxane-containing regimen - Disease must have recurred ≤ 12 months after discontinuation of platinum therapy - Patients who previously received weekly taxane are potentially eligible, provided that they did not progress during therapy or within 3 months of completing therapy - Patients with platinum-refractory disease, as defined by progression during primary or subsequent platinum-based therapy or persistent radiographic disease after primary or subsequent platinum-based therapy, will be included - Patients must have measurable disease in target lesions or assessable disease (defined by CA-125 per protocol), and disease progression per Response Evaluation Criteria in Solid Tumors (RECIST) version 1.1 or modified Gynecologic Cancer Intergroup (GCIG) CA-125 criteria Exclusion Criteria: Patients meeting any of the following exclusion criteria are not to be enrolled in the study: - Prior treatment with an Aurora A-targeted agent (including MLN8237) - Treatment with clinically significant enzyme inducers within 14 days prior to the first dose of MLN8237 and during the study - Treatment with more than 4 cytotoxic chemotherapy regimens in the metastatic setting; prior therapy cannot include more than 2 prior taxane-containing regimens - Known hypersensitivity to Cremophor® EL, paclitaxel or its components - Prior history of ≥ Grade 2 neurotoxicity or any toxicity requiring discontinuation from taxane chemotherapy that is not resolved to ≤ Grade 1 - Comorbid or unresolved toxicity that would preclude administration of weekly paclitaxel - Primary central nervous system malignancy or carcinomatous meningitis - Symptomatic brain metastasis - Inability to swallow oral medications or maintain a fast - History of hemorrhagic or thrombotic cerebrovascular event in past 12 months - Surgery within 3 weeks before study enrollment and not fully recovered - Diagnosis or treatment of another malignancy within 2 years preceding first dose of MLN8237 and have any evidence of residual disease except nonmelanoma skin cancer or in situ malignancy completely resected - Pregnant or lactating - Serious illness that could interfere with protocol completion - Investigational treatment 21 days prior to first dose of MLN8237 - Prior allogeneic bone marrow or organ transplantation - Infection requiring systemic antibiotic therapy within 14 days prior to first dose of MLN8237 - Known history of human immunodeficiency virus (HIV) infection, hepatitis B, or hepatitis C - Radiotherapy to > 25% bone marrow or whole pelvic radiotherapy - Requirement for constant administration of proton pump inhibitor, H2 antagonist, or pancreatic enzymes. Intermittent uses of antacids of H2 antagonists are allowed