Climate change is generating novel communities composed of new combinations of species. These result from different degrees of species adaptations to changing biotic and abiotic conditions, and from differential range shifts of species. To determine whether the responses of organisms are determined by particular species traits and how species interactions and community dynamics are likely to be disrupted is a challenge. Here, we focus on two key traits: body size and ecological specialization. We present theoretical expectations and empirical evidence on how climate change affects these traits within communities. We then explore how these traits predispose species to shift or expand their distribution ranges, and associated changes on community size structure, food web organization and dynamics. We identify three major broad changes: (i) Shift in the distribution of body sizes towards smaller sizes, (ii) dominance of generalized interactions and the loss of specialized interactions, and (iii) changes in the balance of strong and weak interaction strengths in the short term. We finally identify two major uncertainties: (i) whether large-bodied species tend to preferentially shift their ranges more than small-bodied ones, and (ii) how interaction strengths will change in the long term and in the case of newly interacting species.