Paul R. Halmos introduced the concept of subnormality in 1950 at the same time that he introduced hyponormality. Both of these concepts were iired by the unilateral shift, perhaps the best understood non-normal operator. The unilateral shift was the dominant example of a subnormal operator for twenty-five years. on the other hand, there exists a strong connection between operator theory, special subnormal operators, and the moment problem. They interact very often, sometimes in a subtle, unexpected way. It is possible to use a subnormality result, used to solve a moment problem. Conversely, there are situations when the solution to a moment problem leads to the existence of a normal extension for some operators. the present work endeavor to present several results sustaining the interplay mentioned above, as well as the necessary background to understand those phenomena, both is a bounded or an unbounded context. In special case a moment problem is give by following: Let I ? R be an interval. For a positive measure µ on I the nth moment is defined by , provided the integral exists. If (s n ) n?0 is a sequence of real numbers, the moment problem on I consists of solving the following three problems: (I) Dose there exist a positive measure µ on I with moments (s n ) n?0 ? In the affirmative, (II) is this positive measure uniquely determined by the moments (s n ) n?0 ? If not, (III) how can one describe all positive measures on I with moments (s n ) n?0 ? Without loss of generality we may always assume that s 0 = 1. This is just a question of normalizing the involved measures to be probability measures. When µ is a positive measure with moments (s n ) n?0 , we say that µ is a solution to the moment problem. If the solution of moment problem is unique, the moment problem is called determinate. Otherwise the moment problem is side to be indeterminate. We know there are three essentially different types of (closed) intervals. For historical reasons the moment problem on [0, ?) is called the Stieltejes moment problems, the moment problem on R is called Hamburger moment problem, and the moment problem on [0, 1] is referred to as the Hausdorff moment problem. In chapter one we give a history of subnormal operators and moment problems. In chapter two and three we give a brief review of measure theory, operator theory and normal operators. In chapter four we give an explicit solution to the (scalar) moment problem on semi-algebraic compact subsets of R n , and apply this result to the study of some operator multi-sequences. In 1 chapter five we discuss integral representations and extensions of positive functionals in some spaces of fractions of continuous functions. In this process, numerical characterizations for the solvability of power moment problems and the uniqueness of the solutions in unbounded sets will be obtained. As an application, we obtain several necessary and sufficient conditions for the existence of normal extensions for some ltr"