The first part introduces van Kampen diagrams in Cayley graphs of finitely generated, the group is limited, but rather discuss the van Kampen lemma, isoperimetric functions or Dehn functions, the theory of small cancellation groups and the introduction of hyperbolic groups. One of the main tools in geometric group theory is the study of space, in particular geodesic spaces and manifolds, so that the group acted upon. The second part is dedicated to the function toDehn, negatively curved groups, in particular, CAT (0) group, and the complex cubic cubings. In the last section, fill function is presented from the point of geometry, algebra and algorithmic perspective, but it is discussed how filling functions interact, and applications to nilpotent groups, hyperbolic groups and asymptotic cones are given.