Positive semidefinite operator-valued kernel generalizes the well-known notion of reproducing kernel, and is a main concept underlying many kernel-based vector-valued learning algorithms. In this talk I will give a brief introduction to learning with operator-valued kernels, discuss current challenges in the field, and describe convenient schemes to overcome them. I'll overview our recent work on learning with functional data in the case where both attributes and labels are functions. In this setting, a set of rigorously defined infinite-dimensional operator-valued kernels that can be valuably applied when the data are functions is described, and a learning scheme for nonlinear functional data analysis is introduced. The methodology is illustrated through speech and audio signal processing experiments.