In the classical Hardy space on the disk, inner functions play a critical role in operator and function theory. We will discuss inner functions from a general perspective in a large class of Hilbert function spaces and provide an analogue of the notion of finite Blaschke product. Further, we'll understand precisely how these functions relate to socalled Shapiro--Shields functions and shift-invariant subspaces generated by polynomials. Based on joint work with T. Le.