A salient feature of algebraic geometry is the fact that parameter spaces of algebraic objects can often themselves be given the structure of an algebraic scheme. This course aims to make this idea precise, and develop tools to construct such moduli spaces algebraically. We will start with an in-depth look at Grothendieck's classical construction of the Hilbert scheme before systematically studying deformation theory and representability criteria of moduli functors. There will be lots of examples of the constructions of specific moduli spaces, which depending on the interests of the students may include: the Picard scheme, the moduli space of stable curves, moduli spaces of stable sheaves, etc. Further topics may include a brief introduction to algebraic stacks and Artin's representability theorems.