A sequential spectral method (SSM) is a spectral method in which the unknown coefficients of the approximate spectral expansion are computed sequentially, one at a time, instead of simultaneously, like in a classical spectral method (CSM). These methods are especially suited for approximating difficult nonlinear integro-differential equations where the solution behavior is largely unknown and the problem lies in finding a solution of the large scale nonlinear system of equations for the unknown expansion coefficients. The SSM leads by iteration to the same solution as the underlying CSM and thus inherits its spectral accuracy, but it avoids the simultaneous solution of a large system of linear equations. We prove convergence of the SSM to the spectral approximation of the underlying spectral method for a general class of nonlinear elliptic integro-differential equations, and illustrate the effectiveness of the method on two applications.