Abstract: Varieties are the main geometric objects whose properties are studied in algebraic and arithmetic geometry. In particular, there are several interesting invariants of varieties we may consider, like dimension, genus etcetera. To study how varieties, and their invariants, change in families, it can be very advantageous to use moduli spaces. You can think of these as parameter spaces, in which each point corresponds to a variety of a certain type - for example, an elliptic curve, with dimension 1 and genus 1. We will discuss several examples of families of varieties, how we can describe them with a moduli space, and how this can help us to classify, construct and count varieties with prescribed invariants.