Abstract: A Riemannian metric is said to be Einstein if it has constant Ricci curvature. A central problem in differential geometry is to determine which smooth compact manifolds admit an Einstein metric, and to completely understand the moduli space of all such metrics when they exist . The 4-dimensional case of this problem appears to be highly atypical. This lecture will explore the features of 4-dimensional geometry which lead to this remarkable situation, and survey our current knowledge concerning the existence of Einstein metrics on 4-manifolds. Particular emphasis will be placed on 4-manifolds which admit either a complex structure or a symplectic structure, as our current results are particularly sharp in this setting.