This paper deals with functional regression, in which the input attributes as well as the response are functions. To deal with this problem, we develop a functional reproducing kernel Hilbert space approach; here, a kernel is an operator acting on a function and yielding a function. We demonstrate basic properties of these functional RKHS, as well as a representer theorem for this setting; we investigate the construction of kernels; we provide some experimental insight. Nonlinear functional regression: a functional RKHS approach