We give a survey of a series of papers by Birget, Olshanskii, Rips and myself about asymptotic invariants of groups and connections between them and other areas of mathematics. In particular we give an algebraic characterization of groups with word problem in NP, construct manifolds with arbitrary isoperimetric functions $(say f(x)=x^{pi+e})$, and finitely presented counterexamples to von Neumann conjecture (a finitely presented non-amenable group without free subgroups).