Donaldson-Thomas invariants are integers assigned to a smooth, projective threefold X. These integers are defined by a virtual count of points on a Hilbert scheme of curves on X, and are invariant under deformations of X.
We give an introduction to this theory, and explain its relation to the similar curve-counting theory of Gromov-Witten invariants. We present different techniques for calculating Donaldson-Thomas invariants and for generalizing the invariants to nonprojective X. In the last chapter we calculate the Donaldson-Thomas invariants of a threefold X with a map to a surface S having fibres isomorphic to a fixed elliptic curve E.