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An optimum selective transmission scheme for energy-limited sensor networks, where sensors send or forward messages of different importance (priority), is developed. Considering the energy costs, the available battery, the message importances and their statistical distribution, sensors decide whether to transmit or discard a message so that the importance sum of the effectively transmitted messages is maximized. It turns out that the optimal decision is made comparing the message importance with a time-variant threshold. Moreover, the gain of the selective transmission scheme, compared to a nonselective one, critically depends on the energy expenses, among other factors. Albeit suboptimal, practical schemes that operate under less demanding conditions than those for the optimal one are developed. Effort is placed into three directions: 1) the analysis of the optimal transmission policy for several stationary importance distributions; 2) the design of a transmission policy with invariant threshold that entails asymptotic optimality; and 3) the design of an adaptive algorithm that estimates the importance distribution from the actual received (or sensed) messages. Numerical results corroborating our theoretical claims and quantifying the gains of implementing the selective scheme close this paper.