Sirolimus and Azacitidine in Treating Patients With High Risk Myelodysplastic Syndrome or Acute Myeloid Leukemia That is Recurrent or Not Eligible for Intensive Chemotherapy

Inclusion Criteria: 1. Patients must have a diagnosis of one of the following: * MDS (Arm A): High-risk MDS defined as: \>5% blasts in bone marrow and/or the following cytogenetic categories: presence of inv(3)/t(3q)/del(3q), -7/del(7q), complex cytogenetics (3 or more abnormalities) * AML (Arm B): Relapsed/refractory/unable to tolerate conventional chemotherapy * MDS or AML as above BUT with prior therapy with Azacitibine (Arm C): Patients who meet criteria for either Arm A or Arm B but have been treated or are currently treated with Azacitibine \*Note: As of July 2018, only high risk MDS patients will be eligible as Arm B is closed. As of October 2017, those patients with MDS who have received prior treatment will now be enrolled in Arm A as Arm C is closed. 2. Patients must be ≥ 18 years old 3. Patients must have an ECOG performance status of \<= 2 (see Attachment 1). 4. Patients must have a life expectancy of at least 4 weeks. 5. Patients must be able to consume oral medication. 6. Patients must have completed any radiotherapy four weeks prior to study entry, 0-2 weeks for local palliative XRT (small port). 7. Patients must have recovered from the toxic effects of any prior chemotherapy to \< Grade 2 (except for alopecia). 8. Required initial laboratory values: Creatinine≤ 2.0mg/dL; total or direct bilirubin ≤ 1.5mg/dL (if not due to the leukemia itself or known Gilbert's Syndrome);(as documented by treating physician) SGPT(ALT) ≤ 3xULN; glucose \<200 mg/dL, negative pregnancy test for women of child-bearing potential. 9. Patients must be able to sign consent and be willing and able to comply with scheduled visits, treatment plan and laboratory testing. 10. Patients may have had a prior stem cell transplant (autologous or allogeneic), however they may not have active GvHD, nor be on any immunosuppression Exclusion Criteria: 1. Patients must not be receiving any chemotherapy agents (except Hydroxyurea) * Intrathecal ARA-C and intrathecal methotrexate are permissible (as they are not systemic and only isolated to the central nervous system). * Patients can not have received more than 3 prior lines of therapy for their hematologic malignancy. Patient may have previously had azacitidine or decitabine will be eligible to enroll on Arm A (MDS) 2. Patients must not be receiving growth factors. 3. Patients with a current second malignancy requiring systemic therapy, other than non-melanoma skin cancers, are not eligible. If a patient has had a prior second malignancy that is not currently requiring active treatment, the patient will be considered eligible. 4. Patients with uncontrolled high blood pressure, unstable angina, symptomatic congestive heart failure, myocardial infarction within the past 6 months or serious uncontrolled cardiac arrhythmia are not eligible. 5. Patients may not take any of the following medications while on study, but will be considered eligible if medication is discontinued 72 hrs prior to first dose of Sirolimus: * Carbamazepine (e.g. Tegretol) * Rifabutin (e.g. Mycobutin) * Rifampin (e.g. Rifadin) * Rifapentine (e.g. Priftin) * St. John's Wort- may decrease effects of sirolimus by decreasing the amount of sirolimus in the body * Clarithromycin (e.g. Biaxin) * Cyclosporin e.g. (Neoral or Sandimmune) * Diltiazem (e.g. Cardizem) * Erythromycin (e.g. Akne-Mycin, Ery-Tab) * Itraconazole (e.g. Sporanox) * Fluconazole (e.g. Diflucan) * Ketoconazole (e.g. Nizoral) * Telithromycin (e.g. Ketek) * Verapamil (e.g. Calan SR, Isoptin, Verelan) * Voriconazole (e.g. VFEND) - May increase the effects of sirolimus by increasing the amount of this medicine in the body. Can take 72 hours after last dose of Sirolimus * Tacrolimus (e.g. Prograf) - May cause liver transplant rejection or serious side effects in patients on sirolimus. 6. Patients with known HIV positivity or AIDS-related illness are not eligible. 7. Patients with other severe concurrent disease which in the judgment of the investigator would make the patient inappropriate for entry into this study are ineligible. 8. Patients must not have received any investigational agents within 21days of study entry. 9. Patients must not be pregnant or breastfeeding. Pregnancy tests must be obtained for all females of child-bearing potential. Pregnant or lactating patients are ineligible for this study due to the unknown human fetal or teratogenic toxicities of rapamycin. Males or females of reproductive age may not participate unless they have agreed to use an effective contraceptive method. 10. Patients who have uncontrolled infection are not eligible. Patients must have any active infections under control. Fungal disease must be stable for at least 2 weeks before study entry. Patients with bacteremia must have documented negative blood cultures prior to study entry.