In this paper we consider Distance Hedonic Games (DHGs), a class of non-transferable utility coalition formation games that properly generalizes previously existing models, like Social Distance Games (SDGs) and unweighted Fractional Hedonic Games (FHGs). In particular, in DHGs we assume the existence of a scoring vector α, in which the i-th coefficient α _i expresses the extent to which an agent x contributes to the utility of an agent y if they are at distance i. We focus on Nash stable outcomes in the arising games, i.e., on coalition structures in which no agent can unilaterally improve her gain by deviating. We consider two different natural scenarios for the scoring vector, with monotonically increasing and monotonically decreasing coefficients. In both cases we give NP-hardness and inapproximability results on the problems of finding a social optimum and a best Nash stable outcome. Moreover, we characterize the topologies of coalitions that provide high social welfare and consequently give suitable bounds on the Price of Anarchy and on the Price of Stability.