Unlike standard congestion games, weighted congestion games and congestion games with player-specific delay functions do not necessarily possess pure nash equilibria. It is known, however, that there exist pure equilibria for both of these variants in the case of singleton congestion games, i.e., if the players’ strategy spaces contain only sets of cardinality one. In this paper, we investigate how far such a property on the players’ strategy spaces guaranteeing the existence of pure equilibria can be extended. We show that both weighted and player-specific congestion games admit pure equilibria in the case of matroid congestion games, i.e., if the strategy space of each player consists of the bases of a matroid on the set of resources. We also show that the matroid property is the maximal property that guarantees pure equilibria without taking into account how the strategy spaces of different players are interweaved.additionally, our analysis of player-specific matroid congestion games yields a polynomial time algorithm for computing pure equilibria. We also address questions related to the convergence time of such games. For player-specific matroid congestion games, in which the best response dynamics may cycle, we show that from every state there exists a short sequences of better responses to an equilibrium. For weighted matroid congestion games, we present a superpolynomial lower bound on the convergence time of the best response dynamics showing that players do not even converge in pseudopolynomial time.