he present proposal is meant as a four year project for a PhD-student in number theory, in particular Diophantine approximation. One of the basic problems in Diophantine approximation is to investigate, how well a given algebraic number can be approximated by other algebraic numbers. The central result on this problem is Roth's Theorem, which gives an optimal solution. In symmetric Diophantine approximation, one takes two varying algebraic numbers and investigates how well they can approximate each other. The only non-trivial results on this problem, obtained by the PI, are far from optimal. Their proofs are based on a very weak, but proved version of the abc-conjecture. It is possible to develop a theory analogous to Diophantine approximation where one considers approximation by algebraic functions instead of algebraic numbers. In this setting there is a fully proved version of the abc-conjecture. It is likely, that this will lead to symmetric Diophantine appproximation results for algebraic functions much stronger than those obtained for algebraic numbers. The project is about symmetric Diophantine approximation both for algebraic numbers and algebraic functions. A first aim is to prove strong symmetric Diophantine approximation results for algebraic functions. A second aim is to improve some of the existing results for algebraic numbers, and also to formulate reasonable conjectures about what the optimal results should be.