There is a very simple algorithm for the inference of posteriors for probability models on trees. This algorithm, known as Belief Propagation” is widely used in coding theory, in machine learning, in evolutionary inference, among many other areas. The talk will be devoted to the analysis of Belief Propagation in some of the simplest probability models. We will highlight the interplay between Belief Propagation, linear estimators (statistics), the Kesten-Stigum bound (probability) and Replica Symmetry Breaking (statistical physics). We will show how the analysis of Belief Propagation allowed to proof phase transitions for phylogenetic reconstruction in evolutionary biology and develop optimal algorithms for inference of block models. Finally, we will discuss the computational complexity of this simple” algorithm.