In this thesis, we present an expanded account of a computational method for finding the solution of linear time-varying delay systems based on articles by M. Razzaghi and H. R. Marzban (2001) and H. R. Marzban and M. Razzaghi (2004). Also, we extend the proposed method to solve piecewise constant delay systems. For this purpose, the operational matrix of delay corresponding to piecewise constant delay systems is constructed. Dynamical systems with time delays have received much attention in the past few decades, since time delays are frequently encountered in many practical systems, such as communication networks, chemical processes, traortation and power systems. The presence of delays makes analysis and control design much more complicated. Time delay systems are therefore an important calss of systems which their analysis has been of interest to many investigators. It is well known that it is difficult to analytically solve a time-varying delay system with piecewise constant delay function. Several numerical methods have been used to obtain an approximate solution for delay differential equations. Orthogonal functions have been applied to solve various problems of dynamic systems. Much progress has been made towards the solution of delay systems. Signals frequently have mixed features of continuity and jumps. In such situations, neither the continuous basis functions nor piecewise constant basis functions taken alone would from an efficient basi in the representation of such signals. The method is based upon hybrid of block-pulse functions and Chebyshev polynomials. The approach is based on expanding various time functions in the system as their truncated hybrid functions. The associated operational matrices of integration, product and delay are used to reduce the solution of time-delay systems to the solution of algebraic equations. Moreover, we introduce the delay operational matrix of piecewise constant delay systems and apply for solving these systems. This matrix contains many zeros, hence making the approach computationally very attractive. It is also shown that the hybrid of block-pulse functions and Chebyshev polynomials. Illustrative examples are included to demonstrate the validity and applicability of the proposed method. The method is easy to implement and yields very accurate results. 2000 MSC: Primary 49J25, 49K25, 34K35. Secondary 93C05, 93B99. Key words: Chebyshev polynomials, Block-pulse functions, Hybrid functions, Delay operational matrix.