The first part of this problem is about classifying the topology of real projective algebraic curves and surfaces with respect to theirdegree. During the last few decades significant progress has been made on this problem and, as a result, a branch of applied topology relating the topological properties of a real algebraic variety to those of its complexification started to develop. The lecture will focus on the presentation of various geometrical, topological, and some arithmetical tools used in this field. These will be illustrated by their application to the first part of Hilbert's 16th problem and related questions.