The ASCENT Trial is a single arm, multi-center, phase II study. The primary objective is to determine the rejection-free, severe graft-versus-host disease (GVHD)-free survival in pediatric patients with serious non-malignant hematologic diseases (NMHD) undergoing unrelated donor (URD) hematopoietic stem cell transplantation (HSCT) with abatacept added to conventional GVHD prophylaxis. The secondary objective is to characterize the impact of abatacept on infection and the reconstitution of protective immunity to infection. Transplanted patients will be followed for 3 years. Weight-based peripheral blood samples will be drawn longitudinally through two years to evaluate immune reconstitution. The study will enroll 28 pediatric patients with serious NMHD undergoing URD HSCT. The trial will include two strata, based on donor matching. Stratum 1 (n=14) will be for patients with 7/8 donors and stratum 2 (n=14) will be for those with 8/8 (matched) donors. All participants will receive 8 doses of abatacept (10 mg/kg intravenously on days -1, +5, +14, +28, +56, +84, +112, and +150). Recruitment is expected to last for about 2 years and participants will be followed for up to 3 years. This trial will test the hypothesis that extended abatacept administration (combined with a standard regimen of tacrolimus and mycophenolate mofetil) will effectively prevent acute and chronic GVHD in children and adolescents receiving URD HSCT, without compromising their engraftment or reconstitution of protective immunity to infection.