Abstract: We consider several identities involving the multiple harmonic series v^ 1which converge when the exponents /, are at least 1 and i\ > 1.There is a simple relation of these series with products of Riemann zeta functions (the case k = 1) when all the i } exceed 1.There are also two plausible identities concerning these series for integer exponents, which we call the sum and duality conjectures.Both generalize identities first proved by Euler.We give a partial proof of the duality conjecture, which coincides with the sum conjecture in one family of cases.We also prove all cases of the sum and duality conjectures when the sum of the exponents is at most 6. MICHAEL HOFFMANand l'l,Ϊ2,.••,/*)=(so (1) is S(α, ft)).With this notation, S(i) = Λ(ι) = £(/).The relation between the S 's and A's should be clear: for example,Note that (2) implies ,4(2, 1) = C(3).It is immediate from the definitions that 5(11, h)+s(i 2 , ii) = c(ίoc(/2) + c(/i + ω and A(h , I 2 ) +^(/2, I'l) = C(/l)ί(l2) -C(/l + 12) whenever i\, i^> 1.More generally, if ίi, iι, ... , i^ > 1 the sums Σ ^(ίσίl) J » ίσ(ik)) and