Inclusion Criteria:  Patient must be age >= 18 years Patient has a Karnofsky performance status of >= 70% Patient has a life expectancy of >= 3 months Patient has histologically-confirmed, diagnosis of a grade III or IV glioma (including glioblastoma, anaplastic astrocytoma, gliosarcoma, anaplastic oligodendroglioma, anaplastic oligoastrocytoma, not otherwise specified [NOS]), or has a prior, histologically-confirmed, diagnosis of a grade II glioma and now has radiographic findings consistent with a high-grade glioma (grade III or IV) Imaging studies show evidence of recurrent, supratentorial tumor(s). The presence of infratentorial tumor is allowed if the patient also has supratentorial disease that is amenable to placement of an intracavitary Rickham catheter Patient's high-grade glioma has recurred or progressed after prior treatment with brain radiation and temozolomide The patient must be in need of surgery for tumor resection Based on the neurosurgeon's judgment, there is no anticipated physical connection between the post-resection tumor cavity and the cerebral ventricles Absolute neutrophil count (ANC) of >= 1000 cells/mm^3 Platelet count >= 100,000 cells/mm^3 Total bilirubin =< 2.0 mg/dl Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase [SGOT]) =< 4 times the institutional upper limit of normal Serum creatinine =< the institutional upper limit of normal At least 6 weeks must have elapsed since taking a nitrosourea-containing chemotherapy regimen At least 4 weeks since completing a non-nitrosourea-containing cytotoxic chemotherapy regimen (except temozolomide: only an interval of 23 days is required from the last dose administered when patient has been recently treated with the standard temozolomide regimen of daily for 5 days, repeated every 28 days) At least 2 weeks from taking the last dose of a targeted agent At least 4 weeks from the last dose of bevacizumab There is no limit to the number of prior therapies for enrollment during treatment schedule escalation; however, once the maximum tolerated treatment schedule has been identified further enrollment to complete the accrual goal of 12 participants treated at the maximum tolerated treatment schedule will be limited to glioblastoma patients at first or second recurrence All participants must have the ability to understand and the willingness to sign a written informed consent The effects of this treatment on a developing fetus are unknown. Therefore, female patients of childbearing potential and sexually-active male patients must agree to use an effective method of contraception while participating in this study. Women of childbearing potential must have a negative pregnancy test =< 2 week prior to registration  Exclusion Criteria:  Patient has anti-human leukocyte antigen (HLA) antibodies specific for HLA Class I antigens (A*01, A*31, B*07, B*15, C*07) expressed by the neural stem cells Patient is receiving radiation, chemotherapy, or another investigational agent Patient has had prior therapy with neural stem cells Patient has not recovered from any toxicity (> grade 1) of prior therapies, except alopecia Patient is unable to undergo a brain MRI Patient has chronic or active viral infections of the central nervous system (CNS) Patient has a coagulopathy or bleeding disorder Patient has an uncontrolled illness including ongoing or active infection Patient has another active malignancy Patient is pregnant or breastfeeding A patient has a serious medical or psychiatric illness that could, in the investigator's opinion, potentially interfere with the safety monitoring requirements and completion of treatment according to this protocol