In the last years the understanding of planar lattice models has improved dramatically. A framework for the critical large scale limit of such models has been given in the form of SLE (and later CLE), and the convergence to such limits has been proved in some cases. In this context we will discuss certain recent results for percolation type models on planar lattices.
We will then move on to lattices in higher dimensions, where our understanding is much more limited. Nevertheless, in recent years some exciting results appeared, and hopefully others will follow.