The purpose of this paper is to investigate continuous maps from the standpoint of geometric topology and algebraic topology. Using a direct system approach and an inverse system approach of continuous maps, we study the (co)shape and (co)homological properties of continuous maps. Applications of the obtained results include the constructions of long exact sequences of continuous maps for the (co)homology pro-groups and (co)homology inj-groups, spectral Čech (co)homology groups and spectral singular (co)homology groups.