  We develop and study a generalization of commutative rings called bands,
along with the corresponding geometric theory of band schemes. Bands generalize
both hyperrings, in the sense of Krasner, and partial fields in the sense of
Semple and Whittle. They from a ring-like counterpart to the field-like
category of idylls introduced by the first and third author.
  The first part of the paper is dedicated to establishing fundamental
properties of bands analogous to basic facts in commutative algebra. In
particular, we introduce various kinds of ideals in a band and explore their
properties, and we study localization, quotients, limits, and colimits.
  The second part of the paper studies band schemes. After giving the
definition, we present some examples of band schemes, along with basic
properties of band schemes and morphisms thereof, and we describe functors into
some other scheme theories.
  In the third part, we discuss some ``visualizations'' of band schemes, which
are different topological spaces that one can functorially associate to a band
scheme $X$.
