Solitons play a pivotal role in many scientific and engineering phenomena. Solitons are a special kind of nonlinear waves that are able to maintain their shape along the promulgation. From the last four decades, the rampant part of fundamental phenomenon of soliton has successfully attracted the researchers from the physical and mathematical sciences. Various branches of science like solid-state physics, plasma physics, particle physics, biological systems, Bose-Einstein-condensation and nonlinear optics are enjoying the benefits taken from soliton. Soliton research gives way to theoretical aspects such as soliton existence, computation of soliton profiles and soliton stability theory by using the tools of soliton dynamics and soliton interactions to applicative aspects. The hub of this thesis is to search not only for the solitary solutions of nonlinear differential equations but also for nonlinear fractional differential equations. This piece of writing targets to give an intuitive grasp for; Further Improved (G¢ /G) -expansion, Extended Tanh-function, Improved (G¢ /G) -expansion, Alternative (G¢ /G) -expansion with generalized Riccati equation, (G¢ /G, 1/G) -expansion and Novel (G¢ /G) - expansion methods. Moreover, we shall extend Novel (G¢ / G) -expansion method to nonlinear fractional partial differential equations arising in mathematical physics. For multifarious applications, all the methods are glib to follow. In addition, these methods give birth to several types of the solutions like hyperbolic function solutions, trigonometric function solutions and rational solutions. The premeditated methods are very efficient, reliable and accurate in handling a huge number of nonlinear differential equations.