This colloquium lecture will include self-contained introductions to Einstein metrics and Kaehler geometry.  We will see that the behaviorof Einstein metrics indimension 4is quite different from  anything seen in other dimensions. In the process, we will meet some 4-manifolds that admit Einstein metrics, and also make the acquaintance of some others that do not. In particular, we will see  that  the existence  or non-existence of Einstein metrics on compact 4-manifolds strictly depends on diffeomorphism type rather than on homeomorphism type.