The purpose of the following thesis is to study the effects of applying a Value-at-Risk constraint on the classic portfolio optimization problem. The Value at Risk constraint is thoroughly studied and a useful restatement is obtained. The different methods of solving an unconstrained optimization problem are discussed and extended to solve the problem with a VaR constraint. The discussion is then extended to consider the role of information in portfolio optimization problems by solving the problem under partial information. It is shown that the optimal constrained portfolio is a “modification” of the optimal unconstrained portfolio. The results are then further extended to apply in a jump diffusion market. It is shown that the VaR constraint can be evaluated in such markets as well, making it possible to solve the portfolio optimization problem with a VaR constraint in a jump diffusion market.