Morse theory is a powerful mathematical framework for studying the topology of manifolds through the analysis of critical points of smooth real-valued functions. By linking differential analysis with topology, it reveals how local function behavior affects global topological structure. Over the past two decades, Morse theory has played a central role in major developments in random topology, largely due to its inherently local-to-global perspective. In this talk, we will review key results in this area, with a focus on the pivotal contributions of Morse theory. We will also outline some of the main open challenges in the probabilistic Morse-theoretic analysis.