We discuss two notions of percolation on graphs. The first one is $r$-neighbor bootstrap percolation in which an activation process of the vertices of a given graph is carried out. The process starts with some initially activated vertices and then, in each round, any inactive vertex with at least $r$ active neighbors becomes activated. The second one is the so called graph bootstrap percolation in which we activate the edges of a graph instead of vertices. Starting with some initially activated edges, we activate any inactive edge which creates a new copy of $H$, the pattern or base graph. Two problems, one of combinatorial nature and the other of probabilistic type are reviewed. The minimum size of a set of initially activated vertices leading to the activation of all vertices of a graph $G$ in the $r$-neighbor bootstrap percolation is denoted by $m(G, r)$. Computing $m(G, r)$ is a combinatorial problem. In the second problem, each vertex of $G$ is initially activated with a probability $p$ and the question is to find the probability threshold of activation of all vertices of the graph $G$. We talk about the latest developments on both problems and also the connections between $r$-neighbor bootstrap percolation and graph bootstrap percolation.

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