In the Diploma thesis we discuss the basic properties of functions of several variables. We focus on continuity, partial derivatives, Schwarz's theorem and differentiablaty. We present examples that illustrate that continuity and partial differentiation are not merely simple generalizations of analogous terms in the theory of functions of one variable. We also use examples to illustrate Schwarz's theorem. We show that continuity of partial derivatives is a sufficient but not necessary condition for differentiability.