We investigate the relationship between several enumeration complexity classes andfocus in particular on problems having enumeration algorithms with incremental andpolynomial delay ( IncP andDelayP respectively). We show that, for some algorithms, wecan turn an average delay into a worst case delay without increasing the space complexity,suggesting that IncP1 = DelayP even with polynomially bounded space. We use theExponential Time Hypothesis to exhibit a strict hierarchy inside IncP which gives the firstseparation of DelayP andIncP. Finally we relate the uniform generation of solutions toprobabilistic enumeration algorithms with polynomial delay and polynomial space.