In this paper, we describe a Branch-and-Bound (B&B) approach for the problem of minimizing the total tardiness of a set of jobs on a single machine. This is an important objective in several applications where a payment is required in the case of delayed jobs. A new lower bound has been introduced based on several heuristics for the problem. An experimental campaign has been performed showing that our approach can handle up to 20 jobs in a reasonable time.