The main purpose of this study is to evaluate the safety of the study drug known as LY3090106 in healthy participants and in participants with RA who are having an inadequate response to methotrexate (MTX). The study will investigate how the body processes the study drug and how the study drug affects the body. The study will last about 3 months for each participant.

Inclusion Criteria for Healthy Participants:

• Males or non-pregnant females age 18 to 65 years
• Body Mass Index (BMI) of 18.5 to 32.0 kilograms per meter squared (kg/m^2)

Inclusion Criteria for Participants for RA:

• Diagnosis of at least mildly active adult-onset RA
• Positive anti-citrullinated protein (ACPA) antibodies or rheumatoid factor
• Regular use of MTX for at least 12 weeks and at a stable dose (10 to 25 mg/week) for at least 4 weeks prior to baseline
• American College of Rheumatology (ACR) Functional Class I, II or III

Exclusion Criteria for Healthy Participants:

• Have received biologic agents within 3 months or 5 half-lives
• Have surgery within 12 weeks prior to screening
• Known history of human immunodeficiency virus (HIV infection and/or positive human HIV antibodies)
• Have evidence of, or test positive for, hepatitis C virus or hepatitis B virus
• Have evidence of active infection or fever
• Have donated greater than 500 mL blood within 30 days prior to screening
• Smoke greater than 10 cigarettes/day or equivalent
• Have received live vaccine within 1 month of screening
• Have a history of allergy to monoclonal antibodies or severe drug allergies

Exclusion Criteria for Participants with RA:

• Have received prior or current treatment with biologic RA therapies
• Have received a live vaccine 28 days prior to screening or intend to receive a live vaccine during the course of the study
• Hemoglobin < 10 grams per deciliter (g/dL), platelet count < 100,000 cells/microliter (uL), total white blood cell count < 3000 cells/uL, neutrophil count < 2000 cells/uL, or lymphocyte count < 500 cells/uL
• Aspartate transaminase (AST) > 1.5 x upper limit or the normal range (ULN), alanine transamine (ALT) > 1.5 x ULN, creatinine > 1.5 mg/dL (114 micromoles/liter [uMol/L])
• Treatment with > 10 mg/day or unstable dose of oral prednisone or equivalent within 28 days of baseline
• Have required an increase in dose or a new prescription of narcotic within 28 days prior to randomization
• Have received any parenteral corticosteroids within 28 days before screening
• Treatment with any oral disease-modifying anti-rheumatic drugs (DMARDs) other than MTX within 4 weeks prior to baseline
• Any neurological, psychiatric, vascular, or system disorder that could also affect the evaluation of disease activity assessments. Anemia of chronic disease is allowed if hemoglobin > 10 g/dL
• Have rheumatic or systemic autoimmune disease other than RA or significant active systemic involvement secondary to RA. Participants with secondary Sjogren's syndrome are eligible.
• Uncontrolled arterial hypertension characterized by a confirmed systolic blood pressure of > 150 mm Hg or diastolic blood pressure > 90 mm Hg at screening or on day of study drug dosing
• Poorly controlled diabetes as defined by glycosylated hemoglobin (HbA1c) > 8.0% at screening
• Known history of human immunodeficiency virus (HIV infection and/or positive human HIV antibodies)
• Have evidence of, or test positive for, hepatitis C virus or hepatitis B virus