Author: Jan-Erik Roos (Department of Mathematics, Stockholm University, SE-10691 STOCKHOLM, Sweden email: jeroos@matematik.su.se) Title: Theorems and conjectures in homological algebra inspired by computer calculations. Abstract: I will give a survey of some recent results in homological algebra, related to commutative ring theory and algebraic geometry, some of them first found by computer experiments and then proved "by hand". These results cover e.g. the strange homological structure of certain defining ideals of monomial curves (partly joint work with Sturmfels and Fr\"oberg), characterizations of Koszul algebras , structure of the homotopy Lie algebra and also the structure of the (double) Yoneda Ext-algebra of a local commutative noetherian ring etc. The computer programs used are BERGMAN (by J\"orgen Backelin and others), CBAS (by Clas L\"ofwall) and ANICK (by Victor Ufnarovskij, Alexander Podoplelov, J. Backelin and others) as well as the more familiar Macaulay2 (by Dan Grayson and Mike Stillman).