The purpose of this paper is to propose a new branch-and-bound algorithm for a class of scheduling problems to minimize total tardiness on identical parallel machines. In this algorithm, the Lagrangian relaxation technique is applied to obtain a tight lower bound. In addition, the job dominance conditions for the single machine total tardiness problem are utilized for both restricting branches and improving the lower bound. As will be shown by computational experiments, instances with up to 25 jobs and with any number of machines are optimally solved by the proposed algorithm.