Study to Explore the Mechanism of Action of Ocrelizumab and B-Cell Biology in Participants With Relapsing Multiple Sclerosis (RMS) or Primary Progressive Multiple Sclerosis (PPMS)

Inclusion Criteria: General Inclusion Criteria: * For women of childbearing potential: agreement to remain abstinent (refrain from heterosexual intercourse) or use contraceptive methods that result in a failure rate of \<1 percent (%) per year during the treatment period and for at least 24 weeks after the last dose of study treatment or until their B-cells have repleted, whichever is longer Inclusion Criteria Specific to RMS Participants: * Diagnosis of RMS in accordance with the 2010 revised McDonald criteria * Expanded Disability Status Scale (EDSS) score of 0 to 5.5 points, inclusive, at Screening * Disease duration from the onset of multiple sclerosis symptoms less than (\<) 15 years in participants with an EDSS score greater than (\>) 5.0 at Screening * Either treatment-naive or receiving treatment with disease-modifying therapies, including prior use of interferon (IFN)-beta-1a (Avonex®, Rebif®), IFN-beta-1b (Betaseron®/Betaferon), or glatiramer acetate (Copaxone®). * At least one clinically documented relapse in the past year and/or at least one T1-weighted Gadolinium (Gd)-enhancing lesion in the past year and/or at least one new T2 lesion in the past year at the time of enrollment Inclusion Criteria Specific to RMS Cohort Arm 4 Participants: * Must meet inclusion criteria for the RMS cohort * Separate signed Informed Consent Form for the RMS Delayed Time to Start Control Arm (Arm 4) * Must be willing to remain on the same dose and regimen of current standard of care, or no treatment if treatment-naïve, for 12 weeks after study enrollment The treating and/or study physician must agree that the participant is eligible to remain on the same dose and regimen of their current standard of care at Screening, or to receive no treatment if the participant is treatment-naïve, for 12 weeks after study enrollment Inclusion Criteria Specific to PPMS Participants: * Diagnosis of PPMS in accordance with the 2010 revised McDonald criteria * EDSS score of 3.0 - 6.5 points, inclusive, at Screening * Disease duration from the onset of multiple sclerosis symptoms \<10 years in participants with an EDSS at Screening less than or equal to (\</=) 5.0 * Documented history of either elevated immunoglobulin G (IgG) Index or one or more IgG oligoclonal bands (OCBs) detected by isoelectric focusing Exclusion Criteria: * Diagnosis of secondary progressive multiple sclerosis without relapses for at least 1 year * History or known presence of recurrent or chronic infection (e.g., human immunodeficiency virus \[HIV\], syphilis, tuberculosis) * History of recurrent aspiration pneumonia requiring antibiotic therapy * History of cancer, including solid tumors and hematological malignancies (except basal cell, in situ squamous cell carcinomas of the skin, and in situ carcinoma of the cervix of the uterus that have been excised and resolved with documented clean margins on pathology) * History of or currently active primary or secondary immunodeficiency * History of coagulation disorders * History of severe allergic or anaphylactic reactions to humanized or murine monoclonal antibodies * History of alcohol or other drug abuse within 24 weeks prior to enrollment * Known presence or history of other neurologic disorders Significant, uncontrolled disease, such as cardiovascular (including cardiac arrhythmia), pulmonary (including chronic obstructive pulmonary disease), renal, hepatic, endocrine, gastrointestinal, or any other significant disease * Congestive heart failure (according to New York Heart Association III or IV functional severity) * Known active bacterial, viral, fungal, mycobacterial infection, or any major episode of infection requiring hospitalization or treatment with IV antibiotics * Any concomitant disease that may require chronic treatment with systemic corticosteroids or immunosuppressants during the course of the study * Contraindications or intolerance to oral or IV corticosteroids, including IV methylprednisolone, according to the country label * Contraindication for LP * Previous treatment with B cell-targeted therapies (such as rituximab, ocrelizumab, atacicept, belimumab, or ofatumumab) * Previous treatment with natalizumab/Tysabri®, alemtuzumab, anti-CD4 agents, cladribine, teriflunomide, cyclophosphamide, mitoxantrone, azathioprine, mycophenolate mofetil, cyclosporine, methotrexate, total body irradiation, or bone marrow transplantation * Treatment with fingolimod/Gilenya®, dimethyl fumarate/Tecfidera®, or similar treatment within 6 months prior to enrollment * Receipt of a live vaccine within 6 weeks prior to enrollment * Systemic corticosteroid therapy within 4 weeks prior to Baseline * Previous or concurrent treatment with any investigational agent or treatment with any experimental procedure for multiple sclerosis (such as treatment for chronic cerebrospinal venous insufficiency) * Certain laboratory abnormalities or findings at Screening * Inability to complete an MRI * Lack of peripheral venous access * Pregnant or lactating, or intending to become pregnant during the study Exclusion Criteria Specific to RMS Participants: * Diagnosis of PPMS or secondary progressive multiple sclerosis without relapses