  Evolutionary game theory has become one of the most diverse and far reaching
theories in biology. Applications of this theory range from cell dynamics to
social evolution. However, many applications make it clear that inherent
non-linearities of natural systems need to be taken into account. One way of
introducing such non-linearities into evolutionary games is by the inclusion of
multiple players. An example is of social dilemmas, where group benefits could
e.g.\ increase less than linear with the number of cooperators. Such
multiplayer games can be introduced in all the fields where evolutionary game
theory is already well established. However, the inclusion of non-linearities
can help to advance the analysis of systems which are known to be complex, e.g.
in the case of non-Mendelian inheritance. We review the diachronic theory and
applications of multiplayer evolutionary games and present the current state of
the field. Our aim is a summary of the theoretical results from well-mixed
populations in infinite as well as finite populations. We also discuss examples
from three fields where the theory has been successfully applied, ecology,
social sciences and population genetics. In closing, we probe certain future
directions which can be explored using the complexity of multiplayer games
while preserving the promise of simplicity of evolutionary games.
