Study of Osimertinib With and Without Ramucirumab in Locally Advanced or Metastatic Non-Small Cell Lung Cancer (NSCLC)

Inclusion Criteria: * Written informed consent and HIPAA authorization for release of personal health information prior to registration. NOTE: HIPAA authorization may be included in the informed consent or obtained separately. * Age ≥ 18 years at the time of consent. * Histologically or cytologically confirmed non-squamous, non-small cell lung cancer. * Locally advanced or metastatic disease, not amenable to curative surgery or radiotherapy. * Patients must have one of the following: * NSCLC which harbours EGFR Exon 19 deletion. * NSCLC which harbours EGFR L858R mutation. EGFR deletion/mutation must be documented by a Clinical Laboratory Improvement Amendments (CLIA) certified test (either from tissue or ctDNA from blood is allowed). If EGFR deletion/mutation testing has not been done, it should be ordered per standard of care. * Eastern Cooperative Oncology Group (ECOG) performance status 0-1 (Appendix A). * Measurable disease per RECIST 1.1. * Patients with brain metastases are eligible if they are asymptomatic, are treated, or are neurologically stable for at least two weeks without the use of steroids or on stable or decreasing dose of ≤ 10 mg daily prednisone (or equivalent). These criteria must be met on day of consent. * Ability to take pills by mouth. * Previous treatment with cytotoxic chemotherapy or immunotherapy is allowed. * Patients must have adequate hematologic, coagulation, hepatic, and renal function. All laboratory tests must be obtained less than 4 weeks from study entry. This includes: * ANC \>/= 1,500/mm3 * platelet count \>/=100,000/mm3 * HgB ≥ 9 g/dL (may be with transfusion) * Creatinine ≤ 1.5x ULN or creatinine clearance (measured via 24-hour urine collection) ≥40 mL/minute (that is, if serum creatinine is \>1.5 times the ULN, a 24-hour urine collection to calculate creatinine clearance must be performed). * The patient's urinary protein is ≤ 1+ on dipstick or routine urinalysis (UA; if urine dipstick or routine analysis is ≥ 2+, a 24-hour urine collection for protein must demonstrate \< 1000 mg of protein in 24 hours to allow participation in this protocol). * Total Serum Bilirubin ≤ 1.5 x ULN (Patients with known Gilbert Syndrome, a total bilirubin ≤ 3.0 x ULN, with direct bilirubin ≤ 1.5 x ULN) * SGOT, SGPT ≤ 3 X ULN if no liver metastasis present * SGOT, SGPT ≤ 5 X ULN if liver metastasis present * The patient has adequate coagulation function as defined by International Normalized Ratio (INR) ≤ 1.5 and a partial thromboplastin time (PTT) (PTT/aPTT) \< 1.5 x upper limits of normal \[ULN\]. Patients on full-dose anticoagulation must be on a stable dose (minimum duration 14 days) of oral anticoagulant or low molecular weight heparin (LMWH). If receiving warfarin, the patient must have an INR ≤ 3.0. For heparin and LMWH there should be no active bleeding (that is, no bleeding within 14 days prior to first dose of protocol therapy) or pathological condition present that carries a high risk of bleeding (for example, tumor involving major vessels or known varices. * Females of childbearing potential must have a negative serum pregnancy test within 7 days of starting of treatment. See the protocol for additional details. * Females of childbearing potential and Non-sterilized males who are sexually active with a female partner of childbearing potential must agree to use adequate contraception as outlined in the protocol. Exclusion Criteria: * Subjects unable to stop use of medications that are potent inducers of CYP3A4 or known to prolong QT interval. See Appendix B for additional details. * Any prior history of other cancer within the prior 2 years with the exception of: adequately treated basal cell carcinoma, cervical intraepithelial neoplasia \[CIN\]/cervical carcinoma in situ, melanoma in situ or ductal carcinoma in situ \[DCIS\], localized Gleason 6 prostate cancer, papillary thyroid cancer or other non-melanoma skin cancers. * Previous treatment with any EGFR TKIs, including erlotinib, gefitinib, afatinib, avitinib, dacomitinib, rociletinib, or osimertinib. * Previous treatment with any anti-VEGF medications, including vandetinib, nintedanib, bevacizumab, or ramucirumab. * Spinal cord compression unless asymptomatic or stable for at least 2 weeks prior to start of study treatment. * Any unresolved toxicities from prior therapy greater than CTCAE grade 1 (with the exception of alopecia grade 2) at the time of starting study treatment. * Any evidence of severe or uncontrolled systemic diseases, including uncontrolled hypertension and active bleeding diatheses, which in the investigator's opinion makes it undesirable for the subject to participate in the trial or which would jeopardize compliance with the protocol, or active infection including hepatitis B, hepatitis C and human immunodeficiency virus (HIV). Screening for chronic conditions is not required. * Refractory nausea and vomiting, chronic gastrointestinal diseases, inability to swallow the formulated product or previous significant bowel resection that would preclude adequate absorption of osimertinib. * Past medical history of interstitial lung disease (ILD), drug-induced ILD, radiation pneumonitis which required steroid treatment, or any evidence of clinically active interstitial lung disease. * Patients with uncharacterized eye disorders. * History of hypersensitivity of osimertinib or ramucirumab (or active or inactive excipients of osimertinib or ramucirumab). * Judgment by the investigator that the patient should not participate in the study if the patient is unlikely to comply with study procedures, restrictions and requirement. * Any of the following cardiac criteria: * Mean resting corrected QT interval (QTc using Fridericia's formula) \> 470 msec. * Any clinically important abnormalities in rhythm, conduction or morphology of resting ECG e.g., complete left bundle branch block, third degree heart block, second degree heart block, PR interval \>250msec. * Any factors that increase the risk of QTc prolongation or risk of arrhythmic events such as heart failure, hypokalemia, congenital long QT syndrome, family history of long QT syndrome or unexplained sudden death under 40 years of age in first degree relatives or any concomitant medication known to prolong the QT interval. * The patient has experienced any arterial thromboembolic events, including but not limited to myocardial infarction, transient ischemic attack, cerebrovascular accident, or unstable angina, within 6 months prior to first dose of protocol therapy. * The patient has uncontrolled or poorly-controlled hypertension (\>160 mmHg systolic or \> 100 mmHg diastolic for \>4 weeks) despite standard medical management. * The patient has experienced any Grade 3-4 GI bleeding within 3 months prior to first dose of protocol therapy. * The patient has a history of deep vein thrombosis (DVT), pulmonary embolism (PE), or any other significant thromboembolism (venous port or catheter thrombosis or superficial venous thrombosis are not considered "significant") during the 3 months prior to first dose of protocol therapy. * The patient has cirrhosis at a level of Child-Pugh B (or worse) or cirrhosis (any degree) and a history of hepatic encephalopathy or clinically meaningful ascites resulting from cirrhosis. Clinically meaningful ascites is defined as ascites from cirrhosis requiring diuretics or paracentesis. * Any hemoptysis (defined as bright red blood or ≥ 1/2 teaspoon) within 2 months prior to first dose of protocol therapy or with radiographic evidence of intratumor cavitation or has radiologically documented evidence of major blood vessel invasion or encasement by cancer. * The patient has a prior history of GI perforation/fistula (within 6 months of first dose of protocol therapy) or risk factors for perforation. * The patient has a serious or nonhealing wound, ulcer, or bone fracture within 28 days prior to first dose of protocol therapy. * The patient has undergone major surgery within 28 days prior to first dose of protocol therapy, or minor surgery/subcutaneous venous access device placement within 7 days prior to the first dose of protocol therapy. The patient has elective or planned major surgery to be performed during the course of the clinical trial.