This dissertation deals with the application of Estimation Theory in two fields viz. "Estimation of Common Mean" and "Estimation of the Size of a Finite Population". Both the fields already possess vast literature and still attract researchers for further challenging issues. We have tried to address some of these in this thesis which is divided into two parts. In the first four chapters [Part I] we consider the problem of estimation of common mean and in the next two chapters [Part II] we address the problem of estimation of the size of a finite population. All the references cited in the thesis are listed at the end.