We study the role of returns to scale of production function in contests between groups. Two types of group contests are considered, the weakest-link group contest and the best-shot group contest. In each group contest, we investigate the existence and characteristics of the Nash equilibrium of the game, given different returns to scale of the production function for the contest. We find that, in both group contests, the constant returns and decreasing returns to scale bring the same nature of equilibrium. On the other hand, the increasing returns to scale causes the nonconcavity of players' payoff functions, complicates the analysis of the game, and makes changes in equilibrium properties.