  We present a comprehensive construction of scalar, vector and tensor
harmonics on maximally symmetric three-dimensional spaces. Our formalism relies
on the introduction of spin-weighted spherical harmonics and a generalized
helicity basis which, together, are ideal tools to decompose harmonics into
their radial and angular dependencies. We provide a thorough and self-contained
set of expressions and relations for these harmonics. Being general, our
formalism also allows to build harmonics of higher tensor type by recursion
among radial functions, and we collect the complete set of recursive relations
which can be used. While the formalism is readily adapted to computation of CMB
transfer functions, we also collect explicit forms of the radial harmonics
which are needed for other cosmological observables. Finally, we show that in
curved spaces, normal modes cannot be factorized into a local angular
dependence and a unit norm function encoding the orbital dependence of the
harmonics, contrary to previous statements in the literature.
