Nekrasov's derivation of the prepotential of N=2 SUSY gauge theories suggests the existence of a more 'refined' topological string theory partition function. The refined topological vertex is a mathematical tool to compute these partition functions on toric Calabi-Yau 3folds in the A-model. In this talk, first, a short introduction to the topological A-model and to the basic concepts of the toric geometries (which engineer gauge theories) are given. This allows to define the usual topological vertex and its refinement. The derivation of the refined topological vertex is based on a modification of the combinatorial interpr etation of the usual topological vertex. Finally, connections of crystals with symmetries/restrictions and their connection to topological string theory are discussed.