In this thesis, optimal detection performance of centralized and decentralized detection systems is investigated in the presence of cost constrained measurements. For the evaluation of detection performance, Bayesian, Neyman-Pearson and J- divergence criteria are considered. The main goal for the Bayesian criterion is to minimize the probability of error (more generally, the Bayes risk) under a constraint on the total cost of the measurement devices. In the Neyman-Pearson framework, the probability of detection is to be maximized under a given cost constraint. In the distance based criterion, the J-divergence between the distributions of the decision statistics under di erent hypotheses is maximized subject to a total cost constraint. The probability of error expressions are obtained for both centralized and decentralized detection systems, and the optimization problems are proposed for the Bayesian criterion. The probability of detection and probability of false alarm expressions are obtained for the Neyman-Pearson strategy and the optimization problems are presented. In addition, J-divergences for both centralized and decentralized detection systems are calculated and the corresponding optimization problems are formulated. The solutions of these problems indicate how to allocate the cost budget among the measurement devices in order to achieve the optimum performance. Numerical examples are presented to discuss the results.