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Two recent measurements of inclusive jet, 2-jet and 3-jet cross sections indeep-inelastic $ep$ scattering from the H1 collaboration are presented. Themeasurements are performed at low $5<Q^2<100$ GeV$^2$ and high $150<Q^2<15000$GeV$^2$ virtualities of the exchanged boson. The obtained cross sections are ingood agreement with perturbative QCD calculations at NLO and are used toextract the strong coupling $\alpha_s(M_Z)$. The experimental precision is0.6\% and 1.2\% for the high and low $Q^2$ regimes, respectively. The error onthe obtained value of $\alpha_s(M_Z)$ is dominated by the theoreticaluncertainties which are mostly due to missing higher orders. At low $Q^2$ theuncertainty due to terms beyond NLO amounts to 7.5\%. The determination of$\alpha_s$ at various scales shows the running of the strong coupling over alarge range.

Jet Production at Low and High $Q^2$ and Determination of the Strong Coupling $\alpha_s$ at H1