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Abstract We study equilibrium problem on general Riemannian manifolds, focusing on existence of solutions and the convexity properties of the solution set. Our approach consists in relating the equilibrium problem to a suitable variational inequality problem on Riemannian manifolds, and is completely different from previous ones on this topic in the literature. We apply our results to the mixed variational inequality and the Nash equilibrium problem. Moreover, we formulate and analyze the convergence of the proximal point algorithm for the equilibrium problem. In particular, correct proofs are provided for the results claimed in J. Math. Anal. Appl. 388 (2012) 61–77 (i.e., Theorems 3.5 and 4.9 there) regarding the existence of the mixed variational inequality and the domain of the resolvent for the equilibrium problem on Hadamard manifolds. [ABSTRACT FROM AUTHOR]