This lecture, part of the "New trends in Teichmüller theory" program at the International Centre for Theoretical Sciences, explores the geodesic flow on hyperbolic surfaces. Delivered by Ara Basmajian, this first lecture in the series delves into the mathematical foundations of geodesic flows within the context of hyperbolic geometry. Learn about the metric theory of Teichmüller spaces, higher Teichmüller theory, infinite-type surfaces, and connections to anti-de Sitter geometry. Drawing from complex analysis, hyperbolic geometry, dynamics, partial differential equations, and other mathematical disciplines, discover the rich interconnections between different theoretical approaches. Ideal for researchers, graduate students, and mathematicians interested in the cutting-edge developments in Teichmüller theory and its applications to geometric analysis.