Explore the foundations of counting rational points on algebraic varieties in this comprehensive lecture. Delve into the background of this mathematical problem, examining the various phenomena that can occur and reviewing significant proven results. Gain insights into the p-adic Determinant Method and its basic theorem, which will be proven in subsequent lectures. Discover variants of the determinant method and learn how the Approximate Determinant Method is applied to investigate consecutive "powerful" numbers. This 53-minute lecture, presented by Roger Heath-Brown at the Hausdorff Center for Mathematics, serves as the first installment in a series on the Determinant Method, setting the stage for deeper exploration in future sessions.