Summary: We consider weighted congestion games with polynomial latency functions of maximum degree \(d\geq 1\). For these games, we investigate the existence and efficiency of approximate pure Nash equilibria which are obtained through sequences of unilateral improvement moves by the players. By exploiting a simple technique, we firstly show that these games admit an infinite set of \(d\)-approximate potential functions. This implies that there always exists a \(d\)-approximate pure Nash equilibrium which can be reached through any sequence of \(d\)-approximate improvement moves by the players. As a corollary, we also obtain that, under mild assumptions on the structure of the players’ strategies, these games also admit a constant approximate potential function. Secondly, using a simple potential function argument, we are able to show that a \((d+\delta)\)-approximate pure Nash equilibrium of cost at most \((d+1)/(d+\delta)\) times the cost of an optimal state always exists, for every \(\delta\in[0,1]\).