The surreal numbers form a system which includes both the ordinary real numbers and the ordinals. Since their introduction by J. H. Conway, the theory of surreal numbers has seen a rapid development revealing many natural and exciting properties. These notes provide a formal introduction to the theory in a clear and lucid style. The subject is still new and the author is able to lead the reader through to some of the outstanding problems in the field. The topics covered include exponentiation and generalised e-numbers.