  There are many processes in cell biology that can be modeled in terms of
particles diffusing in a two-dimensional (2D) or three-dimensional (3D) bounded
domain $\Omega \subset \R^d$ containing a set of small subdomains or interior
compartments $\calU_j$, $j=1,\ldots,N$ (singularly-perturbed diffusion
problems). The domain $\Omega$ could represent the cell membrane, the cell
cytoplasm, the cell nucleus or the extracellular volume, while an individual
compartment could represent a synapse, a membrane protein cluster, a biological
condensate, or a quorum sensing bacterial cell. In this review we use a
combination of matched asymptotic analysis and Green's function methods to
solve a general type of singular boundary value problems (BVP) in 2D and 3D, in
which an inhomogeneous Robin condition is imposed on each interior boundary
$\partial \calU_j$. This allows us to incorporate a variety of previous studies
of singularly perturbed diffusion problems into a single mathematical modeling
framework. We mainly focus on steady-state solutions and the approach to
steady-state, but also highlight some of the current challenges in dealing with
time-dependent solutions and randomly switching processes
