  Partial wave decomposition is one of the main tools within the modern
S-matrix studies. We present a method to compute partial waves for $2\to2$
scattering of spinning particles in arbitrary spacetime dimension. We identify
partial waves as matrix elements of the rotation group with definite covariance
properties under a subgroup. This allows to use a variety of techniques from
harmonic analysis in order to construct a novel algebra of weight-shifting
operators. All spinning partial waves are generated by the action of these
operators on a set of known scalar seeds. The text is accompanied by a {\it
Mathematica} notebook to automatically generate partial waves. These results
pave the way to a systematic studies of spinning S-matrix bootstrap and
positivity bounds.
