Summary: For each class \(\mathbf A\) of topological spaces we have a closure operation \([ ] : P(X) \to P(X)\), called \(\mathbf A\)-closure, where \(X\) is a topological space and \(P(X)\) is the power set of \(X\). In this paper we study the \(\mathbf A\)-compact spaces, i.e. the topological spaces \((X, \tau) \in {\mathbf A}\) such that the topology \(\tau_{\mathbf A}\) generated by the \(\mathbf A\)-closure is a compact topology. We show that for many classes \(\mathbf A\) the \(\mathbf A\)-compact spaces play the same role in \(\mathbf A\) as the compact Hausdorff spaces play in the class of Hausdorff spaces.