We investigate the potential of a muon collider for testing the presence of anomalous Higgs boson couplings. We consider the case of a light (less than 160 GeV ) Higgs boson and study the effects on the Higgs branching ratios and total width, which could be induced by the non standard couplings created by a class of dim = 6 SU(3) × SU(2) × U(1) gauge invariant operators satisfying the constraints imposed by the present and future hadronic and e − e + colliders. For each operator we give the minimal value of the µ + µ − integrated luminosity needed for the muon collider (µC) to improve these constraints. Depending on the operator and the Higgs mass, this minimal µC luminosity lies between 0.1 f b −1 and 100 f b −1 .One of the main goals of a future µ + µ − Collider (µC), is to provide a Higgs boson factory [1]. It has been shown that with a good energy resolution (of the order of 0.003%) and a reasonable integrated luminosity (of the order of the f b −1 ), the mass and the total width and branching ratios of a light Higgs boson can be directly measured with a high accuracy [2]. In particular, the measurement of the mass and the total width, is the unique feature of a µC [3]. In addition, the branching ratios can be more accurately measured than at a Linear e + e − Collider (LC), as long as m H 160 GeV ; i.e. as long as the total width Γ H is sufficiently small, so that the peak of the µ + µ − → H cross section is enhanced, and a large number of events is produced [2,4].The aim of the present paper is to discuss more precisely the potential of a µ + µ − collider for the search of anomalous Higgs boson couplings. We assume that only a single light (i.e. m H 160 GeV ) standard-like Higgs boson exists. Moreover, we assume that this Higgs boson may have anomalous couplings which can be generated by adding to the Standard Model (SM) lagrangian L SM , a set of new physics (NP) terms associated to a high scale Λ, lying in the several TeV range. These NP terms are expressed in terms of all possible dim = 6 SU(3) × SU(2) × U(1) gauge invariant operators O i involving the various standard model fields and contributing with couplings g i ef f , [5,6,7]. Thus, the contributions of each of these operators on the partial decay widths Γ(H → F ), are determined by g i ef f . Constraints on these coupling constants have already been established from the effects of the aforementioned operators in the gauge sector (at LEP1/SLC, LEP2 and TEVATRON) [8,9,10]. These constraints will be improved by further TEVATRON studies [11], as well as studies at LHC [12] and LC [13] (anticipated to run before the µC), in particular through the direct production of the Higgs boson. For each operator we collect the most stringent constraint on the associated coupling constant g i ef f that should be available by that time. We mention separately the constraints that could be obtained from the study of the Hγγ couplings, if the γγ mode at a Linear Collider will also be available [14,15].Taking the accuracy at which a µC can measure the Higgs total and pa...