Time delay systems have received much attention in the past two decades because time delays are frequently encountered in many practical systems and various fields of engineering and science such as aerospace engineering, robotics, physics, economics, communication networks, chemical processes, traortation systems, industrial processes, population growth, neural networks, climate models, biology, transmission lines and power systems. The presence of delay makes analysis and control design much more complicated. Therefore much effort has been devoted to the analysis, identification and optimal control of various types of time delay systems. In general, they are difficult to analyse and identify. Up to now, a large number of research works have been devoted to the theoretical aspects and numerical treatments of delay differential equations. Models with delay differential equations are usually more complicated than corresponding ordinary differential equations, both with regard to theoretical analysis and numerical simulation. Owing to the lack of smoothness in the associated solutions of delay differential equations, this ltr" basis functions or by piecewise constant basis functions. In recent years, different types of hybrid functions have been successfully applied for solving various problems arising in diverse areas of engineering and science. One of the most advantages of hybrid functions is the good representation of smooth and especially piecewise smooth functions by finite hybrid expansion. In the thesis, a hybrid approximation method is successfully developed to numerically solve piecewise constant delay systems. The first part of this thesis is devoted to the analysis of piecewise constant delay systems. The second part of this thesis is relevant to parameter identification of linear time delay systems with piecewise constant delay function. The method is based on a hybrid of block pulse functions and Taylor polynomials. The operational matrix of delay is constructed. The excellent properties of hybrid functions together with the associated operational matrices of integration, delay and product are then used to transform the main problem into a systems of algebraic equations whose solution is much more easier than the original one. The hybrid of block pulse functions and Taylor polynomials constitutes a semi orthogonal set. The suggested approximation scheme has a simple structure, easy to implement and provides very accurate results. Illustrative examples are included to demonstrate the validity and applicability of the proposed method. The new technique is also applicable to nonlinear piecewise constant delay systems, but some modifications are required.