This richly illustrated text guides readers to discover the beautiful structure of mathematical billiards. Centered around expertly designed problem sets, the book incrementally builds up ideas through threads of related problems, fostering curiosity, exploration, and conversation. Working through the problems, the reader builds an understanding of foundational results, useful techniques, and key examples, all the way up to current research, while hands-on construction activities offer an opportunity to explore phenomena in three dimensions. Beginning with the square billiard table, the opening chapters explore periodic billiard paths in depth, including connections to continued fractions and symbolic dynamics. From this foundation, the book goes on to explore billiards in triangles and other polygons, flows on translation surfaces, interval exchange transformations, outer billiards and tiling billiards, billiards on the ellipse, and many other topics. Throughout, we meet contemporary mathematicians involved in the topics covered, showcasing the research achievements of the vibrant billiards community. Billiards, Surfaces, and Geometry offers a ground level, problem-centered entry point to this exciting field. Basic knowledge of geometry and matrix transformations is assumed, though the main prerequisite is a spirit of curiosity and exploration.