the formulation of the 13th problem in hilbert's address of 1900 to the international congress of mathematicians in paris allows many different interpretations. the most general one was solved by kolmogorov in 1957. however, the more natural "algebraic" form of the problem is still completely open. \vskip .1in \noindent we will describe hilbert's 13th problem in general and will give a algebraic form of it. then we explain some very classical results and a modern approach to them. finally, we shortly introduce a more restrictive form of the 13th problem for which abhyankar has recently given a positive solution.