This paper addresses the problem of scheduling a set of jobs that are released over the time on a set of identical parallel machines, aiming at the minimization of the total weighted completion time. This problem, referred to as P vertical bar r(j)vertical bar Sigma WjCj, is of great importance in practice, because it models a variety of real-life applications. Despite its importance, the P vertical bar r(j)vertical bar Sigma WjCj has not received much attention in the recent literature. In this work, we fill this gap by proposing mixed integer linear programs and a tailored branch-and-price algorithm. Our branch-and-price relies on the decomposition of an arc-flow formulation and on the use of efficient exact and heuristic methods for solving the pricing subproblem. Computational experiments carried out on a set of randomly generated instances prove that the proposed methods can solve to the proven optimality instances with up to 200 jobs and 10 machines, and provide very low gaps for larger instances