  The non-Hermitian skin effect dramatically reshapes the energy bands of
non-Hermitian systems, meaning that the usual Bloch band theory is
fundamentally inadequate as their characterization. The non-Bloch band theory,
in which the concept of Brillouin zone is generalized, has been widely applied
to investigate non-Hermitian systems in one spatial dimension. However, its
generalization to higher dimensions has been challenging. Here, we develop a
formulation of the non-Hermitian skin effect and non-Bloch band theory in
arbitrary spatial dimensions, which is based on a natural geometrical object
known as the amoeba. Our theory provides a general framework for studying
non-Hermitian bands beyond one dimension. Key quantities of non-Hermitian
bands, including the energy spectrum, eigenstates profiles, and the generalized
Brillouin zone, can be efficiently obtained from this approach.
