The main topics of this book are correspondences (dualities) between topological string theories and gauge theories. The first prototypes in this direction are the Gopakumar-Vafa duality between Chern-Simons theory and topological string theory on a special Calabi-Yau manifold called resolved conifold and the Dijkgraaf-Vafa duality between topological string theory and matrix model theory. The first part of the book is devoted to an introduction into gauge theories and their 1/N-expansions. The second part concentrates on topological string theories with an emphasis on non-compact Calabi-Yau manifolds. The third part presents relations between gauge theories and topological string theories as discussed in the first and the second part of the text. The book grew out of graduate level courses given by the author in 2003/2004.