Abstract: The random-sum Poisson-Weibull variable is the sum of a random sample from a Weibull distribution with a sample size that is an independent Poisson random variable. It has a wide range of applications. This random sum is complex and difficult to analyze. Saddlepoint approximations are powerful tools for obtaining accurate expressions for closed-form distribution functions for these complex distributions. The use of saddlepoint approximations almost outperforms other methods with respect to computational costs, though not necessarily with respect to accuracy. This study introduces saddlepoint approximations to the cumulative distribution function for the Poisson-Weibull model, from which we can obtain some important statistical measures of the central tendency of a cumulative distribution. We discuss approximations of a random-sum variable using dependent components, assuming the existence of a moment-generating function. Numerical examples of Poisson-Weibull random sums are presented.