This thesis is concerned with exact solution approaches for the Vehicle Routing Problem. In particular, a detailed review of possible solution methods is given and a new exact solution algorithm for the problem is described. The algorithm we present in this thesis is capable of solving very much larger size (more than double) Vehicle Routing Problems than those previously found in the literature. It is divided into three steps. In the first step, feasible suboptimal routes are implicitly eliminated through the use of bounding and dominance conditions that are derived from State-Space Relaxation techniques. In the second step, feasible routes that were not eliminated in step 1 are explicitly submitted to a series of filter tests that have been developed. As a result, the ones which pass these new tests are included into a subset (hopefully small) of routes, which is guaranteed to contain the optimal Vehicle Routing Problem solution. Finally, in the third step, this optimal solution is obtained through the use of a Set-Partitioning algorithm specialized for the Vehicle Routing Problem. In the computational results that we present, Vehicle Routing Problems involving up to 45 customers were solved exactly. For all the problems tested, at most 950 routes had to be explicitly considered by the Set-Partitioning algorithm. This contrasts with a total of up to a million and a half possible feasible routes for some problems. Finally, as a contribution for future work in the area, a new formulation for the Vehicle Routing Problem (based on two-commodity network flows) is also introduced in this thesis. Although no computational results are presented, a detailed description of an exact solution approach based on this formulation is described.