Explore the fundamental concepts of o-minimality and delve into the intricacies of the Pila-Wilkie Theorem in this 48-minute lecture by Gareth Jones from the University of Manchester. Begin with an introduction to the topic, followed by key definitions and illustrative examples. Examine proofs, the Faffian cell definition, and cell dimension concepts. Investigate the frontier of x and gain insights into the Pila-Wilkie Theorem and Minis Theorem. Consider questions such as "Is it probably bounded?" and "Is there an open box?" Understand the theorem's aim, the introduction of height, and its generality. This talk, part of the Thematic Program on Tame Geometry, Transseries and Applications to Analysis and Geometry, offers a comprehensive exploration of these advanced mathematical concepts.