We discuss the problem how the property "congruence compactness" of an algebra is related to the congruences on this algebra and to the semilattice structure of its CC-congruences (A congruence $\rho$ on an algebra $A$ is called a CC-congruence if the factor algebra $A/\rho$ is congruence compact). Some properties of the (upper) semilattice of CC-congruences of an arbitrary algebra are discussed. The concept of congruence compactness is generalized to subclasses of algebras and congruences. General concepts and results are illustrated by monoids and acts.