Explore advanced asymptotic methods in mathematics through this SISSA/IGAP lecture. Delve into techniques for evaluating infinite sums numerically and recognizing asymptotic patterns in number sequences. Learn both standard approaches like the Euler-Maclaurin formula and less conventional methods, illustrated with diverse examples. Examine challenging problems such as evaluating a slowly convergent sum to high precision, analyzing the asymptotic behavior of complex series coefficients, and computing highly oscillatory infinite series. Gain insights into the circle method, Euler's key point, Hardy-Ramanujan techniques, and generalized asymptotics. Discover transformations, variable changes, and simplification strategies for tackling complex mathematical problems in various branches of mathematics.