We present a review in the construction of accurate and efficient multivariate polynomial approximations on elementary domains that are not Cartesian products of intervals, such as triangles and tetrahedra. After the generalities for high-order nodal interpolation of a function over an interval, we introduce collapsed coordinates and warped tensor product expansions. We then discuss about the choice of interpolation and quadrature points together with the assembling of the final system matrices in the case of elliptic operators. We also present two efficient ways of solving the associated linear system, namely a Schur complement strategy and a p-multigrid solver. Two applications to Computational Fluid Dynamics problems conclude this contribution.