  We consider linear and nonlinear coherent perfect absorbers (CPAs) in
multidimensional geometries and construct explicitly the respective perfectly
absorbed solutions. The multidimensional CPAs have a structure of the so-called
simple layers which represent the generalization of the point $\delta$ function
potential to higher dimensions. The considered examples include broadband CPAs
confined to a straight line (in a two-dimensional setting) and to a plane (in
the three-dimensional space); CPAs for topological vortices on an absorbing
circle; as well as axially-symmetric CPAs on a sphere. Additionally, it is
shown that a paraxial beam propagating along a surface nonlinear CPA embedded
in the three-dimensional space can be stable against perturbations. The results
are interpreted in applications to optical and acoustic systems.
