This is a graduate text on algebraic geometry that provides a quick and fully self-contained development of the fundamentals, including all commutative algebras which are used. A taste of the deeper theory is given: some topics, such as local algebra and ramification theory, are treated in depth. The book culminates with the theory of curves, including the Riemann-Roch theorem, elliptic curves and the zeta function of a curve over a finite field, and the Riemann hypothesis for elliptic curves.