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In this paper we describe the set of conjugacy classes in the group $\mathrm{SL}(n,\mathbb{Z})$. We expand geometric Gauss Reduction Theory that solves the problem for $\mathrm{SL}(2,\mathbb{Z})$ to the multidimensional case, where $\varsigma$-reduced Hessenberg matrices play the role of reduced matrices. Further we find complete invariants of conjugacy classes in $\mathrm{GL}(n,\mathbb{Z})$ in terms of multidimensional Klein-Voronoi continued fractions.