Summary: The aim of this paper is to present a theoretical basis for computing a representation of a compact Riemann surface as an algebraic plane curve and to compute a numerical approximation for its period matrix. We will describe a program CARS [{\it K.-D. Semmler}, {\it T. Ventjoki}, {\it J. Levinen}, {\it L. Renggli} and {\it M. Seppälä}, “CARS. Program for defining Riemann surfaces for computations” (1993-1996)] that can be used to define Riemann surfaces for computations. CARS allows one also to perform the Fenchel-Nielsen twist and other deformations on Riemann surfaces. Almost all theoretical results presented here are well known in classical complex analysis and algebraic geometry. The contribution of the present paper is the design of an algorithm which is based on the classical results and computes first an approximation of a polynomial representing a given compact Riemann surface as a plane algebraic curve and further computes an approximation for a period matrix of this curve. This algorithm thus solves an important problem in the general case. This problem was first solved, in the case of symmetric Riemann surfaces, by {\it M. Seppälä} [Discrete Comput. Geom. 11, No. 1, 65-81 (1994;

Zbl 0805.14029)].