This is a phase II, open label, multi-center, intra-patient dose escalation study to characterize the pharmacokinetics after oral administration of eltrombopag in combination with immunosuppressive therapy in pediatric patients with previously untreated or relapsed/refractory severe aplastic anemia or recurrent aplastic anemia. All patients will be treated with eltrombopag for the 26-week Treatment Period, followed by a 52-week Follow-Up Period. Patients who have been previously untreated with immunosuppressive therapy will be treated according to the standard of care, hATG/cyclosporine, in addition to eltrombopag. Patients with relapsed/refractory SAA or recurrent AA will be enrolled into one of two treatment options: hATG/cyclosporine plus eltrombopag or cyclosporine plus eltrombopag, depending on prior treatment with immunosuppressive therapy. After initiating treatment with eltrombopag, patients will have their dose assessed and modified as tolerated, until the targeted platelet count or maximum dose is achieved. Pharmacokinetic assessments will be performed at time points intended to capture steady state PK of the starting dose and highest dose achieved. There are four separate periods of this study: Screening (signing of written informed consent through Day -1), Treatment (for 26 weeks), Follow-up (additional 52 weeks), and Long-term Follow-up (for additional 3 years). The first 3 periods will be considered the Core phase of the study. Study completion (Core) will occur when the last patient completes the 26-week treatment and 52-week Follow-up Period [at Week 78].