Abstract: The Thompson group F is a countable group of continuous piecewise affine transformations of the unit interval [0,1]. For many years it only existed in unpublished notes of Thompson, but in 1996 Cannon, Floyd and Parry wrote a detailed survey paper on the group F and related goups (D, F', T and V) also due to Thompson. Thompson's group F can be presented as a group with two generators and two relations. Nevertheless it is a rather mysterious group, and a for many years a main open problem about the group is to decide whether it is an amenable group. A recent attempt to prove ameanability of the Thompson group F by E.T. Shavgulidze (Moscow) is currently under investigation. In the seminar talk, I will give a survey of the known results on the Thompson group F and its von Neumann algebra L(F) including also results from a recent joint work with Gabriel Picioroaga "New presentations of Thompson's groups and applications".