We conjecture that every finite groupGhas a short presentation (in terms of generators and relations) in the sense that the totallengthof the relations is (log|G|)O(1).We show that it suffices to prove this conjecture for simple groups.Motivated by applications in computational complexity theory, we conjecture that for finite simple groups, such a short presentation is computable in polynomial time from the standard name ofG, assuming in the case of Lie type simple groups overGF(pm) that an irreducible polynomialfof degreemoverGF(p) and a primitive root ofGF(pm) are given.We verify this (stronger) conjecture for all finite simple groups except for the three families of rank 1 twisted groups: we do not handle the unitary groupsPSU(3,q)=2A2(q), the Suzuki groupsSz(q)=2B2(q), and the Ree groupsR(q)=2G2(q). In particular,all finite groups G without composition factors of these types have presentations of length O((log|G|)3).For groups of Lie type (normal or twisted) of rank≥2, we use a reduced version of the Curtis–Steinberg–Tits presentation.