This paper is concerned with solving some structured multi-linear systems, especially focusing on the equations whose coefficient tensors are M\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {M}$$\end{document}-tensors, or called M\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {M}$$\end{document}-equations for short. We prove that a nonsingular M\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {M}$$\end{document}-equation with a positive right-hand side always has a unique positive solution. Several iterative algorithms are proposed for solving multi-linear nonsingular M\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {M}$$\end{document}-equations, generalizing the classical iterative methods and the Newton method for linear systems. Furthermore, we apply the M\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {M}$$\end{document}-equations to some nonlinear differential equations and the inverse iteration for spectral radii of nonnegative tensors.
