This multicenter, open-label dose-escalation study with an extension phase will evaluate the safety and pharmacokinetics of lumretuzumab in combination with pertuzumab and paclitaxel in participants with metastatic breast cancer expressing HER3 and HER2 protein. Cohorts of participants will receive escalating doses of lumretuzumab intravenously (IV) every three weeks (Q3W) in combination with pertuzumab 840 milligrams (mg) IV initial dose followed by 420 mg IV Q3W and paclitaxel 80 milligrams per square meter (mg/m^2) IV weekly. After completion of dose-limiting toxicity period (21 days), the study will be conducted in two extension phase cohorts: Cohort 1 and Cohort 2. Enrollment in Extension Phase Cohort 2 will occur only upon completion of Extension Phase Cohort 1. Anticipated time on study treatment is until disease progression or unacceptable toxicity occurs.

Inclusion Criteria:

• Histologically confirmed metastatic breast cancer expressing HER3 and HER2 protein
• Participants must be willing to undergo a fresh (pretreatment) tumor/metastases biopsy that will be used to assess the level of HER3 protein expression by immunohistochemistry (IHC) and central pathology review
• HER2 status confirmed on same tumor/metastases by a central laboratory. Breast cancer tumors and/or metastases must be HER2 IHC 1+/in-situ hybridization (ISH)- or HER2 ICH 2+/ISH- as assessed by parallel testing of protein and gene amplification using a Food and Drug Administration (FDA)-approved test
• Eastern Cooperative Oncology Group (ECOG) performance status 0 to 1
• Taxane-naive participants or participants who have received taxanes as part of an adjuvant/neoadjuvant treatment regimen with a disease-free interval of at least 1 year. Participants who have received a docetaxel-containing regimen in the metastatic setting may be eligible. Participants who have received paclitaxel/nab-paclitaxel in the metastatic setting but have discontinued paclitaxel/nab-paclitaxel for a reason other than disease progression and have had a taxane-free interval of at least 6 months may be eligible unless otherwise contraindicated at the investigator's discretion
• Radiologically measurable or clinically evaluable disease according to RECIST criteria
• Last dose of systemic anti-neoplastic therapy greater than (>) 21 days prior to first study treatment infusion. Palliative radiotherapy is allowed up to 2 weeks before the first study treatment infusion
• All acute toxic effects of any prior radiotherapy, chemotherapy or surgical procedure must have resolved to Grade less than or equal to (<=) 1, except alopecia (any grade)
• Adequate hematological, liver and renal function
• Baseline left ventricular ejection fraction (LVEF) of greater than or equal to (>=) 50 percent (%) (measured by echocardiography)
• Female participants of childbearing potential and male participants must agree to use effective contraception as defined by protocol during the study and for at least 6 months after the last dose of study medication
• Participants with Gilbert's Syndrome will be eligible for the study

For extension Phase 2, all of the above except the inclusion criteria mentioned for taxane-naive participants or participants who have received taxanes. In addition, participants in extension Phase 2 may include:

• Participants with no prior chemotherapy for metastatic breast cancer and/or a maximum of only one prior chemotherapy regimen in the adjuvant or neoadjuvant setting
• Taxane-naive participants or participants who have received taxanes as a part of an adjuvant/neoadjuvant treatment regimen with a disease-free interval of at least 1 year

Exclusion Criteria:

• History of clinical evidence of central nervous system (CNS) primary tumors or metastases including leptomeningeal metastases unless they have been previously treated, are asymptomatic and have had no requirement for steroids or enzyme-inducing anti-convulsants in the last 14 days
• Evidence of significant, uncontrolled concomitant diseases which could affect compliance with the protocol or interpretation of results, including uncontrolled diabetes mellitus
• Active or uncontrolled infections
• Known human immunodeficiency virus (HIV) or known active hepatitis B virus (HBV) or hepatitis C virus (HCV) infection
• Major surgery or significant traumatic injury less than (<) 28 days prior to first study treatment infusion (excluding biopsies) or anticipation of the need for a major surgery during study treatment
• Pregnant or breast-feeding women
• Known hypersensitivity to any of the components of RO5479599, pertuzumab or paclitaxel
• Participants with contraindications for paclitaxel therapy according to the Summary of Product Characteristics (SmPC)
• Therapy with an antibody or immunotherapy concurrently or within a period of time where drug exposure is still considered biologically active (usually <5 times t1/2) prior to first dose of study treatment
• Regular immunosuppressive therapy (that is, for organ transplantation, chronic rheumatologic disease)
• Concurrent high doses of systemic corticosteroids (>20 milligrams [mg] of dexamethasone a day or equivalent for >7 consecutive days)
• Baseline QTc interval of >470 milliseconds (ms), participants with baseline resting bradycardia <45 beats per minute or baseline resting tachycardia >100 beats per minute
• Uncontrolled hypertension, unstable angina, congestive heart failure of any New York Heart Association (NYHA) classification, serious cardiac arrhythmia requiring treatment (exceptions: atrial fibrillation, paroxysmal supraventricular tachycardia), history of myocardial infraction within 6 months of enrollment or symptomatic LVEF dysfunction
• A history of Grade >=3 peripheral neuropathy of any etiology