In the reliability theory, the stress- strength models describe the life of a component which has a random strength Y and is subjected to random stress X. The component fails at the instant that the stress applied to it exceeds the strength and the component will function satisfactorily whenever Y X. Thus R=P(X Y) is a measure of component reliability. These models are of substantial interest and usefulness in various subareas of engineering (most prominently, in reliability theory), psychology, genetics, clinical trails, etc. In this thesis we describe the stress- strength models and their applications in different areas and then deal with dir=rtl align=right estimation of R when X and Y are two independent Weibull distributions with different scale parameters, but having the same shape parameter. The maximum likelihood estimator (MLE) and the approximate maximum likelihood estimator of R are studied. We study the approximate distribution of the MLE of R and based on that we obtain asymptotic confidence intervals for R. We also consider the Bayes estimator of R under different prior distributions and finally we use a data analysis for illustrative purposes and apply Mont Carlo simulation for comparing different proposed ltr"