Schubert varieties are known to be Frobenius split in positive characteristic by the work of Mehta and Ramanathan. More recently, Bhatt showed that the full flag variety for GL_n is a splinter by entirely different methods. In this talk, we will explain these concepts and show how to generalize Bhatt's result to all Schubert varieties for all reductive groups. Our methods apply equally well to affine Schubert varieties, which are related to the Langlands program and to the representation theory of algebraic groups. This is joint work with João Lourenço.