An introduction to the arithmetic and geometry of quadratic forms. Topics will include lattices, class number, local-global principle, Gauss composition, and other connections to algebraic number theory. Applications to the representation of integers by quadratic forms will be discussed, with the eventual goal of discussing a proof (joint work with Jonathan Hanke) of Conway's 290-Conjecture: if a positive-definite integral quadratic form takes the values 1, 2, ... , 290, then it takes all (positive) integer values.