In this paper we consider the problem of the disposition of the set of points of a nonsingular real algebraic curve of given degree in ℂP2. The homotopy description of the complement of such a curve in ℂP2 is the first step toward solving the problem of disposition mentioned. In the case of an arbitrary curve we are able to prove that the complement indicated is homotopy equivalent with a three-dimensional cell complex of special form. For a certain class of curves the complex turns out to be two-dimensional and admits a precise description, which allows us to calculate its fundamental group. The results are given without proof. © 1984 Plenum Publishing Corporation.