This chapter provides a theoretical analysis of reproducing kernel Hilbert spaces. It starts by showing that every Hilbert space of functions in which point evaluations are continuous linear functionals possesses a reproducing kernel. It proceeds by showing that every positive semidefinite kernel gives rise to a reproducing kernel Hilbert space—this is the Moore--Aronszajn theorem. Finally, the Mercer theorem offers an explicit representation of this reproducing kernel Hilbert space under additional conditions on the kernel.