During the past few years, several percolation models with long (infinite) range dependencies were introduced. Among them Random Interlacements (introduced by A.-S. Sznitman) and Coordinate Percolation (introduced by P. Winkler). During the talk I will focus on the connectivity properties of these models. The latter model has polynomial decay in sub-critical and super-critical regime in dimension 3. I will explain the nature of this phenomenon and why it is difficult to handle these models technically. In the second half of the talk I will present key ideas of the multi-scale analysis which allows to reach some conclusions. At the end of the talk I will discuss applications and several open problems.