  Mesoscopic particles immersed in a critical fluid experience long-range
Casimir forces due to critical fluctuations. Using field theoretical methods,
we investigate the Casimir interaction between two spherical particles and
between a single particle and a planar boundary of the fluid. We exploit the
conformal symmetry at the critical point to map both cases onto a highly
symmetric geometry where the fluid is bounded by two concentric spheres with
radii R_- and R_+. In this geometry the singular part of the free energy F only
depends upon the ratio R_-/R_+, and the stress tensor, which we use to
calculate F, has a particularly simple form. Different boundary conditions
(surface universality classes) are considered, which either break or preserve
the order-parameter symmetry. We also consider profiles of thermodynamic
densities in the presence of two spheres. Explicit results are presented for an
ordinary critical point to leading order in epsilon=4-d and, in the case of
preserved symmetry, for the Gaussian model in arbitrary spatial dimension d.
Fundamental short-distance properties, such as profile behavior near a surface
or the behavior if a sphere has a `small' radius, are discussed and verified.
The relevance for colloidal solutions is pointed out.
