A space X is said to be cellular-countably compact if for each cellular family U in X, there is a countably compact subspace K of X such that U ∩ K ∅ for each U ∈ U. The class of cellular-countably compact spaces contain the classes of countably compact spaces and cellular-compact spaces and contained in a class of pseudocompact spaces. We give an example of Tychonoff DCCC space which is not cellular-countably compact. By using Erdȍs and Radó's theorem, we establish the cardinal inequalities for cellular-countably compact spaces. We show that the cardinality of a normal cellular-countably compact space with a G δ-diagonal is at most c. Finally, we study the topological behavior of cellular-countably compact spaces on subspaces and products.