Muon collisions at multi-TeV center of mass energies are the ideal place to study Higgs boson properties. At these energies the high production rates and the clean environment will allow precise measurements of its couplings to fermions and bosons. In addition, the double Higgs boson production rate could be sufficiently high to directly measure the trilinear self-couplings strength, giving access to the determination of the Higgs field potential.
This proceeding gives an overview of the results that have been obtained so far on Higgs couplings by studying the $\mu$$^{+}$$\mu$$^{-}$ $\to$ H$\nu$ $\bar{\nu}$ and $\mu$$^{+}$$\mu$$^{-}$ $\to$ HH$\nu$ $\bar{\nu}$ processes. All the studies have been performed by simulating the signal and physics background samples and by evaluating the effects of the beam-induced background on the detector performances. The
evaluations on Higgs boson couplings sensitivities and the most recent results on the uncertainty on double Higgs production cross section, together with the trilinear self-coupling, will be discussed at $\sqrt{s}=$ 3 TeV.