-  Inclusion Criteria

               1. Patients must have histologically or cytologically confirmed epithelial ovarian,
                  fallopian tube, or primary peritoneal carcinoma. Histologic documentation (via
                  the pathology report) of the original primary tumor is required.

               2. Patients must have measurable disease, according to RECIST v1.1.

               3. Patients must have recurrent, platinum-resistant disease (defined as having
                  relapsed within 6 months of last platinum-containing regimen) or be unable to
                  receive further platinum therapy. There is no limit on the number of prior
                  treatment regimens; however, patients may not have previously received weekly
                  paclitaxel in the recurrent setting. Previous dose dense paclitaxel as initial
                  therapy is allowable.

               4. Patients must have the ability to take oral medications.

               5. Females, age ≥18 years.

               6. ECOG performance status ≤2 (Karnofsky ≥60%, see Appendix A).

               7. Life expectancy of greater than 3 months.

               8. Patients must have normal organ and marrow function.

               9. The effects of fostamatinib on the developing human fetus are unknown. For this
                  reason, women of childbearing potential 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 she is participating in this study, she should
                  inform her treating physician immediately.

              10. Patients who are willing and able to comply with the protocol and study
                  procedures including willingness to undergo tumor biopsy or paracentesis for
                  tumor cells before therapy (at baseline) and after initiation of treatment
                  (before Cycle 2) for all subjects if this is clinically and safely feasible to do
                  so.

              11. Patients with known history or current symptoms of cardiac disease, or history of
                  treatment with cardiotoxic agents, should have a clinical risk assessment of
                  cardiac function using the New York Heart Association Functional Classification.
                  To be eligible for this trial, patients should be class 2B or better.

              12. Patients with a prior or concurrent malignancy whose natural history or treatment
                  does not have the potential to interfere with the safety or efficacy assessment
                  of the investigational regimen are eligible for this trial, with permission of
                  the protocol chair.

              13. Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral
                  therapy with undetectable viral load within 6 months are eligible for this trial
                  if the anti-retroviral therapy is not an excluded concurrent medication.

              14. For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV
                  viral load must be undetectable on suppressive therapy, if indicated and the
                  suppressive therapy is not an excluded concurrent medication.

              15. Patients with a history of hepatitis C virus (HCV) infection must have been
                  treated and cured. For patients with HCV infection who are currently on
                  treatment, they are eligible if they have an undetectable HCV viral load and the
                  HCV therapy is not an excluded concurrent medication.

              16. Patients with treated brain metastases are eligible if follow-up brain imaging
                  after central nervous system (CNS)-directed therapy shows no evidence of
                  progression.

          -  Exclusion Criteria

               1. Patients who have had chemotherapy or radiotherapy 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 3 weeks
                  earlier. Hormonal therapy directed at the malignant tumor must be discontinued at
                  least one week prior to registration.

               2. Patients who are currently receiving or have previously received any other
                  investigational agents within 3 weeks prior to entering the study.

               3. Patients with known untreated brain metastases, as progressive neurologic
                  dysfunction may develop that would confound the evaluation of neurologic and
                  other adverse events.

               4. Patients with Grade 2 or greater neuropathy.

               5. History of allergic reactions attributed to compounds of similar chemical or
                  biologic composition to fostamatinib or paclitaxel. Patients who are able to
                  tolerate paclitaxel on a desensitization protocol will be allowed.

               6. Strong CYP3A4 inhibitors or inducers should not be used within 3 days of Day 1
                  dosing until the end of study. Moderate CYP3A4 inhibitors or inducers should be
                  used with caution.

               7. Uncontrolled intercurrent illness

               8. Pregnant women are excluded from this study because the potential for teratogenic
                  or abortifacient effects of fostamatinib are unknown. Because there is an unknown
                  but potential risk for adverse events in nursing infants secondary to treatment
                  of the mother with fostamatinib, breastfeeding should be discontinued if the
                  mother is treated with fostamatinib. These potential risks may also apply to
                  other agents used in this study.