  In this paper it is stressed that there is no {\em physical} reason for
symmetries to be linear and that Lie group theory is therefore too restrictive.
We illustrate this with some simple examples. Then we give a readable review on
the theory finite $W$-algebras, which is an important class of non-linear
symmetries. In particular, we discuss both the classical and quantum theory and
elaborate on several aspects of their representation theory. Some new results
are presented. These include finite $W$ coadjoint orbits, real forms and
unitary representation of finite $W$-algebras and Poincare-Birkhoff-Witt
theorems for finite $W$-algebras. Also we present some new finite $W$-algebras
that are not related to $sl(2)$ embeddings. At the end of the paper we
investigate how one could construct physical theories, for example gauge field
theories, that are based on non-linear algebras.
