Explore the concept of rigid compact complex manifolds in this 50-minute conference talk by Ingrid Bauer from Bayreuth University. Delve into various notions of rigidity in compact complex manifolds, examining their relationships and addressing open questions and conjectures. Focus on product-quotient varieties, which are quotients of compact Riemann surface products under finite group actions. Discover how these structures yield rigid manifolds with intriguing properties. Learn about examples of rigid but not infinitesimally rigid surfaces of general type, providing an answer to a long-standing Morrow-Kodaira question. Examine recent findings on rigid product-quotient varieties of dimension three or higher with Kodaira dimensions 0 and 1. Cover topics such as infinitesimal rigidity, compact complex surfaces, local deformations, and hyperelliptic manifolds throughout the presentation.