An Einstein metric is by definition a Riemannian metric of constant Ricci curvature. One central problem of modern differential geometry is to completely determine which smooth compact $n$-manifolds admit such metrics. In this talk, I will describe ways in which the 4-dimensional case involves phenomena which are utterly unlike anything seen in other dimensions. I will then go on to discuss recent results regarding 4-dimensional Einstein manifolds. Many of these results focus on 4-manifolds that also happen to carry either a complex structure or a symplectic structure.