We introduce and analyze q-potential games and q-congestion games, where q is a positive integer. A 1-potential (congestion) game is a potential (congestion) game. We show that a game is a q-potential game if and only if it is (up to an isomorphism) a q-congestion game. As a corollary, we derive the result that every game in strategic form is a q-congestion game for some q. It is further shown that every q-congestion game is isomorphic to a q-network game, where the network environment is defined by a directed graph with one origin and one destination. Finally we discuss our main agenda: The issue of representing q-congestion games with non-negative cost functions by congestion models with non-negative and monotonic facility cost functions. We provide some initial results in this regard.