If H is a space of functions defined on some set X, evaluating functions at those points provides additional structure to the space H. If the evaluation functionals are bounded on H, we call the overall structure a Reproducing Kernel Hilbert Space (RKHS). This viewpoint goes back to the early XX century, and it was axiomatized in the early 50's by Bergman and Aronszsajn. In this seminar lecture we want to lay down the basics of the theory as in the seminal article od Aronszajn and provide some examples. In forthcoming seminar, we will explore problems of interpolation in RKHS.