We consider a d-parameter Hermite process with Hurst index H = (H-1, . . , H-d) is an element of (1/2, 1)(d) and we study its limit behavior in distribution when the Hurst parameters H-i, i = 1, . . , d (or a part of them) converge to 1/2 and/or 1. The limit obtained is Gaussian (when at least one parameter tends to 1/2) and non-Gaussian (when at least one-parameter tends to 1 and none converges to 1/2). (C) 2018 Elsevier B.V. All rights reserved.