Author’s abstract: Chebyshev polynomials of the first and the second kind in \(n\) variables \(z_ 1,z_ 2,\dots,z_ n\) are introduced. These variables are the characters of the representations of \(SL(n+1,\mathbb{C})\) corresponding to the fundamental weights. The classical Chebyshev polynomials \((\cos kt\) and \(\sin(k+1)t/\sin t\), respectively, as polynomials in \(z=2\cos t)\) are obtained for \(n=1\). The Chebyshev polynomials are eigenpolynomials of a second order linear partial differential operator, which is in fact the radial part of the Laplace- Beltrami operator on certain symmetric spaces. We give an explicit expression of this operator in the coordinates \(z_ 1,z_ 2,\dots,z_ n\), and then show how many results in the literature on differential equations satisfied by Chebyshev polynomals in several variables follow immediately from well-known results on the radial part of the Laplace- Beltrami operator. Related topics like orthogonality, symmetry relations, generating functions and recurrence relations are also discussed. Finally we note that the Chebyshev polynomials are a special case of a more general class of orthogonal polynomials in several variables.