Inclusion Criteria:

          -  Patients must have histologically or cytologically confirmed stage IV (American Joint
             Committee on Cancer [AJCC] 7th Edition) or recurrent non-small cell lung cancer
             (NSCLC)

          -  Patients must have documented presence of an EGFR exon 19 deltion or exon 21 (L858R)
             substitution mutation; T790M mutation or other molecular abnormality will be allowed
             as long as it accompanies one of the mutations listed above; EGFR testing must be
             performed using a Food and Drug Administration (FDA)-approved test or in a Clinical
             Laboratory Improvement Amendments (CLIA)-certified laboratory.

          -  Patients must have tissue available and must agree to submission of tissue and blood;
             one to two paraffin-embedded tissue blocks or 15-20 unstained slides are requested (a
             minimum of 12 slides is required); cytology (i.e. fine-needle aspirations, pleural
             effusion specimens) is acceptable if a cell block or sufficient unstained slides are
             available; tumor material must be reviewed by a local pathologist who must confirm
             that at least 100 viable tumor cells are present in the sample and sign the S1403
             Pathology Review Form; patients must also be willing to submit blood samples for
             correlative research at baseline, during treatment and at progression

          -  Patients enrolled at sites participating in the Repeat Biopsy Study must agree to
             submission of tissue obtained by a repeat biopsy performed at the time of disease
             progression

          -  Patients must not have received any prior systemic anticancer therapy for advanced or
             metastatic disease including chemotherapy or EGFR tyrosine kinase inhibitor therapy
             (including gefitinib, erlotinib, afatinib, or any experimental EGFR tyrosine kinase
             inhibitors [TKI] agents); prior chemotherapy for non-metastatic disease (i.e. adjuvant
             therapy or concurrent chemo-radiotherapy) is allowed as long as > 12 months has passed
             since completion of therapy; adjuvant EGFR-directed therapy is not allowed; local
             therapy (i.e. palliative radiotherapy) is allowed as long as a period of 7 days has
             passed since the last dose was received and the patient has recovered from any
             associated toxicity at the time of registration

          -  Patients may have measurable or non-measurable disease documented by computed
             tomography (CT) or magnetic resonance imaging (MRI) within 42 days prior to
             registration; the CT from a combined positron emission tomography (PET)/CT may be used
             only if it is of diagnostic quality; laboratory parameters are not acceptable as the
             only evidence of disease; in order to qualify as measurable, measurable disease must
             be outside previous radiation field; all disease must be assessed and documented on
             the Baseline Tumor Assessment Form (Response Evaluation Criteria in Solid Tumors
             [RECIST] 1.1)

          -  Patients must have a CT or MRI scan of the brain to evaluate for central nervous
             system (CNS) disease within 42 days prior to registration; patient must not have
             symptomatic brain metastases or evidence of leptomeningeal carcinomatosis; patients
             with asymptomatic brain metastases are eligible if off of steroids for at least 7 days
             prior to registration without development of symptoms

          -  Patients must not have any known clinically active interstitial lung disease

          -  Absolute neutrophil count (ANC) >= 1,500/mcL

          -  Platelets >= 75,000/mcL

          -  Hemoglobin >= 9 g/dL

          -  Total bilirubin =< 1.5 x institutional upper limit of normal (IULN)

          -  Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =< 2.5 x IULN (or
             =< 5 x IULN for patients with known liver metastases)

          -  Serum creatinine =< 1.5 x IULN OR measured or calculated creatinine clearance >= 60
             mL/min

          -  Patients must not have significant gastrointestinal disorders with diarrhea as a major
             symptom (e.g. Crohn's disease, malabsorption, etc)

          -  Patients must be able to swallow medication by oral route

          -  Patients must not have a history of clinically relevant cardiovascular abnormalities
             such as uncontrolled hypertension, congestive heart failure New York Heart Association
             (NYHA) classification of 3, unstable angina or poorly controlled arrhythmia or
             myocardial infarction within 6 months prior to registration; if clinically indicated,
             echocardiogram or multigated acquisition (MUGA) must be performed and cardiac ejection
             fraction must be >= 50%

          -  Patients must not have had major surgery within 28 days prior to registration or be
             scheduled for surgery during the projected course of protocol treatment; tumor biopsy
             is allowed

          -  Patients must not have a known history of active hepatitis B infection (defined as
             presence of hepatitis B surface antigen [Hep B sAg] and/ or Hep B deoxyribonucleic
             acid [DNA]), active hepatitis C infection (defined as presence of hepatitis C [Hep C]
             ribonucleic acid [RNA]) and/or known human immunodeficiency virus (HIV) seropositive

          -  Patients must not have any other concomitant serious illness or organ system
             dysfunction which in the opinion of the investigator would either compromise patient
             safety or interfere with the evaluation of the safety of the study drug

          -  Patients must not be planning to receive any other investigational agents during the
             course of protocol treatment

          -  Patients must not have a history of allergic reactions attributed to compounds of
             similar chemical or biologic composition to afatinib and/or cetuximab

          -  Prestudy history and physical must be obtained with 28 days prior to registration

          -  Patients must have Zubrod performance status of 0 - 2

          -  No other prior malignancy is allowed except for the following: adequately treated
             basal cell or squamous cell skin cancer, in situ cervical cancer, adequately treated
             stage I or II cancer from which the patient is currently in complete remission, or any
             other cancer from which the patient has been disease free for three years

          -  Patients must not be pregnant or nursing; women/men of reproductive potential must
             have agreed to use an effective contraceptive method; a woman is considered to be of
             "reproductive potential" if she has had menses at any time in the preceding 12
             consecutive months; in addition to routine contraceptive methods, "effective
             contraception" also includes heterosexual celibacy and surgery intended to prevent
             pregnancy (or with a side-effect of pregnancy prevention) defined as a hysterectomy,
             bilateral oophorectomy or bilateral tubal ligation; however, if at any point a
             previously celibate patient chooses to become heterosexually active during the time
             period for use of contraceptive measures outlined in the protocol, he/she is
             responsible for beginning contraceptive measures

          -  Patients must be informed of the investigational nature of this study and must sign
             and give written informed consent in accordance with institutional and federal
             guidelines

          -  As a part of the Oncology Patient Enrollment Network (OPEN) registration process the
             treating institution's identity is provided in order to ensure that the current
             (within 365 days) date of institutional review board approval for this study has been
             entered in the system