Evolutionary dynamics can be strongly affected by spatial structure – even in the case of constant fitness values. A popular way to model such spatial structure are networks, where each node represents an individual and offspring can be placed to other sites via the links. How does such a network affect the probability of a new mutation to take over the entire population? And how does it alter the time this process is expected to take? Of particular interest are structures that increase the probability that advantageous mutations take over. Such amplifiers are surprisingly abundant among all networks. However, they do not necessarily maximize the fitness under long term evolution. Applying these models to real populations typically requires that each node is a subpopulation – leading to a different class of models. Also, in the long run the conventional probability of fixation is insufficient to describe the dynamics and one has to take into account the process of continuously generated mutations.