Abstract. Cohomology operations (including the cohomology ring) of a geometric object are finer algebraic invariants than the homology of it. In the literature, there exist various algorithms for computing the homology groups of simplicial complexes (\cite{Mun84}, \cite{DE95,ELZ00}, \cite{DG98}), but concerning the algorithmic treatment of cohomology operations, very little is known. In this paper, we establish a version of the incremental algorithm for computing homology given in \cite{ELZ00}, which saves algebraic information, allowing us the computation of the cup product and the effective evaluation of the primary and secondary cohomology operations on the cohomology of a finite simplicial complex. The efficient combinatorial descriptions at cochain level of cohomology operations developed in \cite{GR99,GR99a} are essential ingredients in our method. We study the computational complexity of these processes and a program in Mathematica for cohomology computations is presented.