Inclusion Criteria:

* Patients must have histologically or cytologically confirmed epithelial ovarian cancer, fallopian tube cancer, or primary peritoneal cancer that has recurred or is refractory to initial therapy; patients must have received platinum-based chemotherapy before entry into this protocol
* Patients must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded) as > 20 mm with conventional techniques or as > 10 mm with spiral computed tomography (CT) scan OR patients must have evidence of progression based on an elevated CA-125 (defined as a value of > 2 x upper limit of normal \[ULN\] documented on two separate determinations made > 2 weeks apart) if the physical exam is normal and CT scan of the chest/abdomen/pelvis, has a disease volume < 1 cm in maximum diameter
* Patients may have received no more than one prior chemotherapy regimen (i.e. initial first-line chemotherapy only)
* Life expectancy of greater than 12 weeks
* Eastern Cooperative Oncology Group (ECOG) performance status =< 2 (Karnofsky >= 60%)
* Leukocytes >= 3,000/mcL
* Absolute neutrophil count >= 1,500/mcL
* Platelets >= 100,000/mcL
* Hemoglobin >= 8 g/dL
* Total bilirubin within normal institutional limits
* Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =< 2.5 × institutional upper limit of normal
* Creatinine within normal institutional limits OR creatinine clearance >= 60 mL/min/1.73 m\^2 for patients with creatinine levels above institutional normal
* Women of child-bearing potential must have a negative pregnancy test prior to study entry; women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation; should a woman become pregnant or suspect she is pregnant while participating in this study, she should inform her treating physician immediately
* Ability to understand and the willingness to sign a written informed consent document

Exclusion Criteria:

* Patients who have had chemotherapy, radiotherapy, or major surgery within 4 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study or those who have not recovered from adverse events due to agents administered more than 4 weeks earlier
* Patients with borderline tumors or tumors of low malignant potential
* Patients with current bowel obstruction
* Patients may not be receiving any other investigational agents nor have participated in an investigational trial within the past 30 days
* Patients with known brain metastases should be excluded from this clinical trial
* History of allergic reactions attributed to compounds of similar chemical or biologic composition to AZD2171 (cediranib maleate)
* Mean corrected QT (QTc) > 470 msec (with Bazett's correction) in screening electrocardiogram or history of familial long QT syndrome
* Greater than +1 proteinuria on two consecutive dipsticks taken no less than 1 week apart
* Uncontrolled intercurrent illness including, but not limited to hypertension, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements
* Pregnant women are excluded from this study, breastfeeding should be discontinued if the mother is treated with AZD2171
* Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral therapy are ineligible
* Any significant abnormality noted in the electrocardiogram (ECG) within 14 days of treatment
* A New York Heart Association classification of III or IV (NOTE: patients classified as class II controlled with treatment may continue with increase monitoring)
* Conditions requiring concurrent use of drugs or biologics with proarrythmic potential; these drugs are prohibited during studies with AZD2171