We study approximate and analytical solutions of nonlinear integral equations, Integro-Differential equations and singular integral equations of Abel's type. The approximate methodswe have used are ADM, noise terms phenomenon in the decomposition series, the modified decomposition method in which the approximate solution converges to the exact solution at a faster rate and the series solution method. The exact methods we have used are direct computation method and the Laplace transform method. We have also employed a combination of Laplace and ADM for some nonlinear integral equations. We have also studied Cauchy's principal value of an integral and properties of Hilbert transform.