Course description: This is a standard introductory course the goal of which is to get acquainted with the basic notions of the field. Thus we start with point-set topology and the study of topological spaces, in particular metric spaces, continuity, connectedness, compactness. The machinery developed will allow us to consider one of the major theorems of topology: the classification of compact, connected surfaces. In the second half of the course we get a taste of algebraic topology - the notion of the fundamental group of a topological space will be introduced as well as several ways of computing it. Throughout the course we will study numerous examples and applications.