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We propose a simple and accurate approximation of the outage probability at the output of L-branch diversity combining receivers with arbitrarily fading channels. The method is based on the saddlepoint approximation, which only requires the knowledge of the moment generating functions of the signal-to-noise ratio at the output of each diversity branch. In many particular cases of practical interest, the outage approximation is obtained in closed-form expressions. Numerical results illustrate the accuracy of the proposed method for practical outage values in a variety of fading scenarios.