In this paper, we consider Legendre multi-Galerkin methods to solve Fredholm integral equations of the second kind with weakly singular kernel and the corresponding eigenvalue problem. We obtain the convergence rates for the approximated solution and iterated solution in weakly singular Fredholm integral equations of the second kind in both L2 and infinity-norm. We also establish error bounds of approximated eigenelements with exact eigenelements in the eigenvalue problem of a compact integral operator with weakly singular kernel in both infinity and L2-norm. Numerical examples are presented to prove the theoretical estimates. © 2018 Elsevier B.V.