Rank-k cartesian-tensor spherical harmonics are defined recursively by the Clebsch-Gordan coupling of rank-(k - 1) tensor spherical harmonics with certain complex basis vectors. By taking the rank-0 tensor harmonics to be the usual scalar spherical harmonics, the new definition generates rank-1 harmonics equivalent to the vector spherical harmonics commonly employed in the quantum theory of angular momentum. A second application of the definition generates new rank-2 harmonics which are orthogonal transformations of the symmetric and antisymmetric rank-2 harmonics defined by Zerilli (1970). Continued application of the definition generates new rank-k harmonics which are orthogonally related to tensors used by Burridge (1966). The main advantage of the new tensor harmonics is that the numerous standard properties (for example, completeness; orthogonality; gradient, divergence and curl formulae; addition formulae) of scalar and vector spherical harmonics, generalize, essentially unchanged in form, to the rank-k case. Furthermore, the recursive definition allows systematic evaluation of integrals of products of three tensor harmonics in terms of Wigner coefficients, the latter immediately implying selection rules and symmetries for the integrals. Together, these generalized properties and coupling integrals permit straightforward spherical harmonic analysis of many partial differential equations in mathematical physics. Application of the new harmonics is demonstrated by analysis of the tensor equations of Laplace and Helmhotz, stress-strain equations for free vibrations of an elastic sphere, the Euler and Navier-Stokes equations for a rotating fluid, and the magnetic induction and mean-field magnetic-induction equations for a conducting fluid. Finally, the method of Orszag (1970) for the fast computation of spherical harmonic coefficients of nonlinear interactions is generalized for the tensor-harmonic case.