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This thesis sets out to compare five well known design rules for the design of model reference adaptive systems. These are the M.I.T. rule, the Liapunov synthesis, the gradient rules of Dressier and Price, and the Monopoli design rule. A systematic performance comparison is made using two low order gain adjustment systems simulated on a digital computer. Step, sinusoidal and stochastic input signals are used and the system state variables and performance criteria are all expressed as dimensionless quantities. The results clearly demonstrate the superior performance of the Liapunov and Monopoli designs. The main disadvantage of other designs is that the dimensionless performance criteria is not a monotonic decreasing function of the dimensionless gain parameter. An analysis of the noisy case is then performed and this further points out the flexibility of the Liapunov synthesis. The next objective of the research is to extend the scope of application of the Liapunov designs. First a modification of the usual design algorithm for multivariable systems is made sc that a wider class of plants, in which the adjustable parameters may appear simultaneously in two or more elements of the plant and control matrices, can be readily treated. Examples are given to illustrate the design procedures and the typical performance of such designs. Secondly, the simultaneous parameter and state estimation system using model reference methods is investigated. Landau's hyperstability design, which can be shown to be equivalent to the Liapunov design, is preferred for this problem. To distinguish this design from the well known Generalized Equation Error (G.E.E.) design, we have called it the Stable Response Error (S.R.E.) design. The practical difficulty of using this globally stable design rule is found to be the implementation of the series (derivative) compensator. It is then shown how the problem is solved by using the state variable filters. Various simulation results substantiate the characteristics (namely unbiased estimates and very fast convergence) of the resulting design. The recovery of the simultaneous state estimates when the state variable filters are used with the S.R.E. design is then considered. With a moderate rate of convergence, the quality of the state estimates is found to be good. The main disadvantage of the S.R.E. method is that the range of parameter variations must be known a priori in order to design the series compensator which ensures the global stability. Finally, the extensions of the S.R.E. method to treat nonlinear and multivariable systems are presented. The main effort here is to find the appropriate structures of the estimation model. To conclude the thesis, a real case study is presented. This is the modelling of a nonlinear, third order internal combustion engine by a linear, first order model. The parameters of the model are adjusted according to the S.R.E. design rule. The practical results obtained demonstrate the feasibility of using the model reference method in a real physical system. Then some of the experiments are repeated with the estimation system based on the G.E.E. design rule. The results are found much inferior to those of the S.R.E. design.