  In his PhD thesis, Einstein derived an explicit first-order expansion for the
effective viscosity of a Stokes fluid with a suspension of small rigid
particles at low density. His formal derivation relied on two implicit
assumptions: (i) there is a scale separation between the size of the particles
and the observation scale; and (ii) at first order, dilute particles do not
interact with one another. In mathematical terms, the first assumption amounts
to the validity of a homogenization result defining the effective viscosity
tensor, which is now well understood. Next, the second assumption allowed
Einstein to approximate this effective viscosity at low density by considering
particles as being isolated. The rigorous justification is, in fact, quite
subtle as the effective viscosity is a nonlinear nonlocal function of the
ensemble of particles and as hydrodynamic interactions have borderline
integrability. In the present memoir, we establish Einstein's effective
viscosity formula in the most general setting. In addition, we pursue the
low-density expansion to arbitrary order in form of a cluster expansion, where
the summation of hydrodynamic interactions crucially requires suitable
renormalizations. In particular, we justify a celebrated result by Batchelor
and Green on the second-order correction and we explicitly describe all
higher-order renormalizations for the first time. In some specific settings, we
further address the summability of the whole cluster expansion. Our approach
relies on a combination of combinatorial arguments, variational analysis,
elliptic regularity, probability theory, and diagrammatic integration methods.
