In this paper, Legendre spectral projection methods are applied for the Volterra integral equations of second kind with a smooth kernel. We prove that the approximate solutions of the Legendre Galerkin and Legendre collocation methods converge to the exact solution with the order O(n-r) in L2-norm and order O(n-r+12) in infinity norm, and the iterated Legendre Galerkin solution converges with the order O(n-2r) in both L2-norm and infinity norm, whereas the iterated Legendre collocation solution converges with the order O(n-r) in both L2-norm and infinity norm, n being the highest degree of Legendre polynomials employed in the approximation and r being the smoothness of the kernels. We have also considered multi-Galerkin method and its iterated version, and prove that the iterated multi-Galerkin solution converges with the order O(n-3r) in both infinity and L2 norm. Numerical examples are given to illustrate the theoretical results. © 2017, SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional.