In this talk we describe {\it metastability and droplet growth} for the two-dimensional lattice gas with Kawasaki dynamics at low temperature and low density. Particles perform simple exclusion on $\mathbb Z^2$, and inside a {\it finite} but arbitrary box $\Lambda_0$ there is a binding energy $U$ between neighboring occupied sites. The initial configuration is chosen such that $\Lambda_0$ is empty while outside $\Lambda_0$ particles are placed randomly with density $\rho = e^{-\beta\Delta}$ for some $\Delta \in (U,2U)$, where $\beta$ is the inverse temperature. Since in equilibrium $\Lambda_0$ wants to be fully occupied when $\beta$ is large, the dynamics will tend to fill $\Lambda_0$ with particles. However, since the particle density $\rho$ is small this will take a long time to happen. We investigate how the transition from empty to full takes place. In particular, we identify the size and shape of the {\it critical droplet}, the time of its creation, and the typical trajectory prior to its creation, in the limit as $\beta \to \infty$. The choice $\Delta \in (U,2U)$ corresponds to the situation where the gas is supersaturated and the critical droplet has side length $>1$.

Because particles are {\it conserved} under Kawasaki dynamics, the analysis of metastability and droplet growth is more difficult than for Ising spins under Glauber dynamics. The key point is to show that at low density the gas outside $\Lambda_0$ can be treated as a reservoir that creates and annihilates particles randomly at the boundary of $\Lambda_0$. Once this has been achieved, the problem reduces to local metastable behavior on $\Lambda_0$, and standard techniques from non-conservative dynamics can be applied. Inside $\Lambda_0$ the dynamics is still conservative, but this can be handled via local geometric arguments.