Why do we want to count them?

It is quite likely that almost everyone who has dabbled in polyomino and animal enumeration has been asked by a friend or relative (or even asked themselves):

Question 9   Why count polyominoes and animals?

In the recent past the mathematics of counting, enumerative combinatorics, has undergone something of a renaissance; arguably the main force behind this has been the role of combinatorics in many problems found at the interface between mathematics and other sciences -- particularly statistical physics, computer science and theoretical chemistry. Many different types of animals and polyominoes can be found in abundance at this interface.

To give the flavour of the sorts of models and problems in which animals and polyominoes can be found, we will describe how animal enumeration problems can be found in the theory of integer partitions, and in the mathematical treatment of magnets, polymers and random media.