This survey is devoted to some nonstandard types of orthogonal polynomials in the complex plane. Under suitable integra,bility conditions on $w$, we consider polynomials orthogonal on a circular arc with respect to a non-Hermitian complex inner product as well as Geronimus' version of orthogonality on a contour in the complex plane. Also, we introduce a class of polynomials orthogonal on some selected radial rays in the complex plane. In both of cases we investigate their existence and uniqueness, recurrence relations, representations and connections with standard polynomials orthogonal on the real line. We also give an introduction to the general theory of orthogonality on the real line and the unit circle. Zero distributions of nonstandard types of orthogonal polynomials are considered.