A Safety Extension Study of Trastuzumab Emtansine in Participants Previously Treated With Trastuzumab Emtansine Alone or in Combination With Other Anti-Cancer Therapy in One of the Parent Studies

Inclusion Criteria: * Completed single-agent trastuzumab emtansine or combination trastuzumab emtansine treatment in the parent study or who continue to receive single-agent trastuzumab emtansine or combination trastuzumab emtansine treatment at the time of the parent study closure and received the last study drug dose within the 6 weeks (42 days) prior to the first dose of study therapy on the extension study or Continue to receive treatment in the control arm of study BO21976/TDM4450g (NCT00679341) at the time of the parent study closure if the participant received the last dose of control arm study drug within the 6 weeks (42 days) prior to the first dose of control arm study therapy in the extension study * Participants in the control arm from Study BO21976/TDM4450g whose disease progression has occurred during the transition interval between the parent study and this extension study may initiate trastuzumab emtansine treatment at the time of enrollment into study TDM4529g (NCT00781612) * Expectation by the investigator that the participant may continue to benefit from additional single-agent trastuzumab emtansine or combination trastuzumab emtansine treatment or Expectation of the investigator that the participant may continue to benefit from control arm treatment as given in study BO21976/TDM4450g and at the time of disease progression may benefit from single-agent trastuzumab emtansine treatment * Women of childbearing potential and men with partners of childbearing potential, must be willing to use a highly effective form of non-hormonal contraception or two effective forms of non-hormonal contraception by the participants and/or partner, and to continue the use of contraception for the duration of study treatment and for at least 5 months after the final dose of atezolizumab (if applicable) or 7 months after the final dose of trastuzumab, trastuzumab emtansine or pertuzumab, whichever is later. Women must refrain from donating eggs during this same period * Male participants whose partners are pregnant should use condoms for the duration of the pregnancy. Men must refrain from donating sperm during this same period Exclusion Criteria: * AEs leading to single-agent trastuzumab emtansine or combination trastuzumab emtansine treatment discontinuation in the parent study * Ongoing SAEs from the parent study * Progressive disease on single-agent trastuzumab emtansine or a trastuzumab emtansine-containing regimen during the parent study or before starting the extension study, with the exception of participants from study TDM4688g (NCT00943670) with early disease progression who went on to receive pertuzumab + trastuzumab emtansine treatment and have not experienced further disease progression on the combination regimen * Peripheral neuropathy of Grade greater than or equal to (\>/=) 3 per the National Cancer Institute Common Terminology Criteria for Adverse Events (NCI CTCAE), Version 3.0, 4.0 or 5.0, as utilized in the parent study * History of symptomatic congestive heart failure (\[CHF\]; New York Heart Association \[NYHA\] Classes II-IV), ventricular arrhythmia requiring treatment, current unstable angina, or history of myocardial infarction within 6 months prior to study entry * Severe dyspnea at rest due to complications of advanced malignancy or current requirement for continuous oxygen therapy * Current severe, uncontrolled systemic disease (for example \[e.g.\] clinically significant cardiovascular, pulmonary, or metabolic disease) * Major surgical procedure or significant traumatic injury within 28 days prior to study entry or anticipation of the need for major surgery during the course of study treatment * Current pregnancy or lactation * History of receiving any investigational treatment or other systemic therapy directed at controlling cancer (e.g., chemotherapy, trastuzumab, etc.) since the participant's last study drug dose in the parent study * History of hypersensitivity with previous trastuzumab emtansine or any agent used with trastuzumab emtansine in the parent study, precluding further dosing * Assessed by the investigator to be unable or unwilling to comply with the requirements of the protocol