In this thesis, Lie - Trotter splitting, Strang - Marchuk splitting and symmetrically weighted sequential (SWS) splitting methods which are known as classical operator splitting methods are considered to find the numerical solution of the various ordinary differential equations (ODEs) and partial dierential equations (PDEs). We also presented their error analyses in order to show advantages and disadvantages of these methods. Firstly, we considered simple linear and nonlinear ODE examples to motivate for the classical operator splitting methods. Then, two numerical examples which consist of a kinetic model of phage infection and the Newell - Whitehead - Segel (NWS) equation are studied. All these examples show that the operator splitting methods are a powerful technique with respect to the accuracy and robustness.