  A number of swimming microorganisms such as ciliates ($\textit{Opalina}$) and
multicellular colonies of flagellates ($\textit{Volvox}$) are approximately
spherical in shape and swim using beating arrays of cilia or short flagella
covering their surfaces. Their physical actuation on the fluid may be
mathematically modeled as the generation of surface velocities on a continuous
spherical surface -- a model known in the literature as squirming, which has
been used to address various aspects of the biological physics of locomotion.
Previous analyses of squirming assumed axisymmetric fluid motion and hence
restricted all swimming kinematics to take place along a line. In this paper we
generalize squirming to three spatial dimensions. We derive analytically the
flow field surrounding a spherical squirmer with arbitrary surface motion, and
use it to derive its three-dimensional translational and rotational swimming
kinematics. We then use our results to physically interpret the flow field
induced by the swimmer in terms of fundamental flow singularities up to terms
decaying spatially as $\sim 1/r^3$. Our results will enable to develop new
models in biological physics, in particular in the area of hydrodynamic
interactions and collective locomotion.
