Our endeavour is to study fractional calculus from a different angle, starting from elementary to higher level of research. This is an entirely new topic different from the traditional system of mathematics. It started from the work of Leibniz in the 17th century but remained untouched till the beginning of the 20th century. The development of the subject started seriously by the researchers at the end of the last quarter of the 20th century, that is why fractional calculus is called calculus of the 20th century. The theory and application of fractional calculus are now tested in the areas of Science,Engineering, Technology, Rheology,Viscoelasticity, Chemical Physics, Electromagnetic Waves,Acoustics,Finance, Statistics and Economics.The physical problems involved in the study of fractional differential equations are both linear and nonlinear in nature. The solution of the types of problems appeared in the discussion of the fractional differential equations needs new types of methods. For this reason new methods like Adomian Decomposition Method, Homotopy Analysis Method,Variational Itaration Method,Generalized Differential Transform Method etc.have been applied in this connection.