Traditional graph partitioning algorithms compute a k-way partitioning of a graph such that the number of edges that are cut by the partitioning is minimized and each partition has an equal number of vertices. The task of minimizing the edge-cut can be considered as the objective and the requirement that the partitions will be of the same size can be considered as the constraint. In this paper we extend the partitioning problem by incorporating an arbitrary number of balancing constraints. In our formulation, a vector of weights is assigned to each vertex, and the goal is to produce a k-way partitioning such that the partitioning satisfies a balancing constraint associated with each weight, while attempting to minimize the edge-cut. Applications of this multi-constraint graph partitioning problem include parallel solution of multi-physics and multi-phase computations, that underlay many existing and emerging large-scale scientific simulations. We present new multi-constraint graph partitioning algorithms that are based on the multilevel graph partitioning paradigm. Our work focuses on developing new types of heuristics for coarsening, initial partitioning, and refinement that are capable of successfully handling multiple constraints. We experimentally evaluate the effectiveness of our multi-constraint partitioners on a variety of synthetically generated problems.