This course aims to introduce the foundations of modern algebraic geometry. These foundations were laid out by Gröthendieck in 1950’s and 60’s and in particular were based on the notion of a scheme. Cohomology of coherent sheaves on schemes can be used to prove many important and fundamental geometric results, furthermore schemes provide a unifying framework for algebraic geometry and number theory. In this course we will carefully define and study various aspects of schemes. Cohomology of coherent sheaves will be the subject of Math 552.