For a smooth projective variety X, the derived category is a natural object to study, containing the data of complexes of (quasi-)coherent sheaves with morphisms being maps of complexes up to a weak notion of equivalence. This turns out to be not only a natural bookkeeping device with which to define derived functors in algebraic geometry and commutative algebra, but an interesting geometric invariant in its own right. I will present the definition of the derived category in this setting and some preliminary results, before describing how they classify curves up to isomorphism with some explanation of how the theory extends to higher-dimensional settings. Some exposure to homological algebra and (quasi-)coherent will be useful, but not necessary. Speaker(s): Saket Shah (Michigan)