Accounts of empathy distinguish between cognitive (attribution of mental states to a social target) and emotional (sharing of emotions with a social target) empathy. To date, however, little is known about whether and how (interactions between) person perceptions, situational characteristics, and the observer-target relationship affect these constructs. The current study hence investigated (a) how the perceived warmth and competence of different social targets relate to both types of empathy, (b) whether there are differences in empathic responding to positive vs. negative scenarios, and (c) the impact of identification with the social targets. Eighty-nine participants rated cognitive and emotional empathy regarding four stereotypical target characters (student, elderly person, businessperson, alcoholic person) facing diverse positive and negative events. They also rated how warm and competent these characters appeared to them and how strongly they identified with the social targets. Results for cognitive and emotional empathy were partly overlapping, but demonstrated several significant differences, thereby demonstrating the need to investigate the two concepts separately. Notably, stereotypes of warmth predicted both cognitive and emotional empathic responses more strongly in desirable than in undesirable scenarios, which may relate to greater freedom of response to positive (rather than negative) social outcomes permitted by society. Our data show that scenario valence mattered even more for cognitive (than for emotional) empathy because it additionally moderated the effects of perceived competence and social identification. Finally, both cognitive and emotional empathy increased as a positive function of social identification, and social identification moderated effects exerted by perceived warmth and competence (yet differently for the two types of empathy investigated). Together, these findings speak to empathic responses arising from a complex interplay between perceptions (i.e., warmth and competence), scenario valence, and social identification.