  In 1982 Lou van den Dries showed that the fact that the field of real numbers is o-minimal gives a large amount of useful information about the definable sets of higher dimension, such as the family of definable subsets of the real plane. Soon after this, Julia Knight, Anand Pillay and Charles Steinhorn noticed that if a structure A is o-minimal, then so is any structure elementarily equivalent to A, and that van den Dries’ analysis of higher-dimensional definable set applies to all these structures. These results led to much activity on the frontier between model theory and function theory. Several old problems from model theory and function theory were solved. Alex Wilkie showed that the field of real numbers with a symbol for exponentiation is o-minimal and has a model-complete complete theory, and thereby gave a positive answer to Tarski’s old problem of whether this structure allows a quantifier elimination, though his method was very far from the syntactic analysis that Tarski had in mind.   (This is one case where we need to remember the difference  between being model-complete and having quantifier elimination;   see section 2 above.  The question whether this particular theory  has quantifier elimination is much harder and is closely related to  a deep conjecture of number theory called Schanuel’s conjecture;   see the paper of Macintyre and Wilkie.) We now know a wide range of ways of adding interesting features to the field of real numbers in such a way that the resulting structure is still o-minimal (and hence in some sense mathematically tractable). Van den Dries has urged that o-minimal structures provide a good setting for developing the ‘tame topology’ programme of Alexander Grothendieck.