Roommate problems might not have a stable solution. But for such problems we are still faced with the problem of matching the agents. One natural approach would be to match the agents in such a way that the resulting matching is “close” to being stable. Such solution concepts should select stable matchings when they exists and select matchings “close” to being stable when the problem does not have any stable matchings. We work with the following solution concepts, Almost Stability, Maximum Irreversibility, Maximum Internal Stability, P-stability and Q-stability, and define a new solution concept, called Iterated P-stability. We investigate consistency, population monotonicity, competition sensitivity and resource sensitivity of these solution concepts. We also explore Maskin monotonicity of these solution concepts.