This thesis is based on some selected topics related to the Alday-Gaiotto-Tachikawa (AGT) duality, vortex counting and topological vertex. It begins with a review on necessary background materials for later chapters. Then we will study the nonabelian vortex counting problem and its relation with the strip amplitudes of topological vertex. After that we will demonstrate a degeneration phenomenon of instanton partition functions of quiver gauge theories and obtain the two-dimensional CFT dual of nonabelian vortices. These results will be generalized to instanton/vortex on orbifolds and the N=1 super Liouville theories in the following chapter.
