This thesis is concerned with aspects of the numerical solution of parabolic integrodifferential equations and it consists of two parts. The first part is concerned with project preliminaries. The second part is concerned with the central theme of this thesis — the numerical solution of parabolic integro-differential equations. The first part of the thesis ( chapters 2 and 3 ) deals with aspects of background knowledge in Numerical Analysis, with emphasis on the numerical solutions of ordinary differential equations (ODEs for short) and efficient numerical solution techniques for systems of one-dimensional linear parabolic partial differential equations (PDEs for short). In particular, we use both comprehensive and sophisticated mathematical software packages and libraries to get the most reliable, robust and efficient numerical routines for solving ODEs and PDEs. In the second part of the thesis, some numerical methods for the solution of integrodifferential equations of parabolic type are discussed, with emphasis on the methods which use time discretization schemes based on the Backward Euler and the Crank- Nicolson schemes. The integral term is approximated in each case by a quadrature rule with relatively high-order truncation error, so that a relatively large time step can be used for the quadrature so as to limit the storage requirements, without sacrificing the overall order of convergence. We describe certain aspects of the numerical algorithms proposed in Le Roux and Thomee [15] and Zhang [20] and, in particular, we examine ways in which the algorithms can be implemented efficiently. The solution algorithm proposed in Zhang [20], referred to in what follows as Modified Method I, is implemented and applied to solve a number of test problems. Based closely on the ideas of Le Roux and Thomee [15] , we construct a second package, referred to in what follows as Modified Method II, which implements a collection of 6 quadrature schemes (the Rectangular Rule, Trapezoidal Rules A, B and I, and Simpson’s Rules A and B). We then test the effectiveness of both packages in terms of improvements in accuracy, storage requirements and execution times by solving some integro-differential equations of parabolic type and analyzing the results. The improved methods reduce greatly both the memory and computational expense involved in solving integro-differential equations of parabolic type. Modified Method I is shown to be very robust and efficient when solving the standard test problems. Modified Method II is more efficient than Modified Method I when solving the same type of problems.