The aim of this book is to present an extensive study of q-homotopy analysis transform method to obtaining approximated analytical solution for complex nonlinear problems arises in the connected areas of science and technology. The intention is to examine the accuracy of proposed technique and its efficiency for solving nonlinear complex fractional differential equations and coupled system of differential equations. This work also attempt to make subject popular and acceptable to the engineering and science communities to appreciate the universe of wonderful mathematics with fractional order differential equations using essential tools.