A rational surface is an algebraic surface birationally isomorphic to \( {\mathbb P}^{2} \). The author investigates the computational problems of rational parametrization of algebraic surfaces and formulates several algorithms for the purpose. He describes preliminary techniques (quadratic surfaces, inversion of birational maps, parametrization of a generic fibre, rational normal curves, Del Pezzo surfaces, adjoints and computation of adjoints). One of the main results of the paper is an algorithm ParameterizeWithAdjoints which takes as input a homogeneous polynomial \( F \in k[x,y,z,w] \), representing a rational surface (previously tested by the algorithm DecideRationality), and outputs a quadruple of bivariate polynomials, representing a rational parametrization of the surface. Let \( q(S) \) be the number of holomorphic 1-forms on the algebraic surface \( S \) over the algebraically closed ground field \( k \) of characteristic zero (irregularity of \( S \)). Let \( P_{2}(S) \) be the plurigenus of \( S \). By the Castelnuovo-Enriques theorem [see \textit{P. Griffiths} and \textit{J. Harris}, ``Principles of algebraic geometry'' (1978; Zbl 0408.14001; 2nd ed. 1994; Zbl 0836.14001); Chapter IV]: If \( S \) is an algebraic surface over an algebraically closed ground field of characteristic zero with \( q(S) = P_{2}(S) = 0 \) than \( S \) is rational. Let \( p_{a} \) be the arithmetic genus of \( S \), and let \( p_{g} \) be the geometric genus of \( S \). Then \( q(S) = p_{g} - p_{a} \). Let \( P_{2}(S) = 0 \). Then \( p_{g} = 0 \). Therefore by \( q(S) = 0 \) and \( P_{2}(S) = 0 \) we have \( p_{a} = 0 \). The author uses a variant of Castelnuovo's theorem with \( p_{a}(S) = P_{2}(S) = 0 \). The appendix of the paper contains a correctness proof of the mentioned parametrization algorithm. The proof is based on the theory of divisors on smooth surfaces over \( k \) and on elements of sheaf theory.