  This article is a survey of conjectures and results on reductive algebraic
groups having good reduction at a suitable set of discrete valuations of the
base field. Until recently, this subject has received relatively little
attention, but now it appears to be developing into one of the central topics
in the emerging arithmetic theory of (linear) algebraic groups over
higher-dimensional fields. The focus of this article is on the Main Conjecture
asserting the finiteness of the number of isomorphism classes of forms of a
given reductive group over a finitely generated field that have good reduction
at a divisorial set of places of the field. Various connections between this
conjecture and other problems in the theory of algebraic groups (such as the
analysis of the global-to-local map in Galois cohomology, the genus problem,
etc.) are discussed in detail. The article also includes a brief review of the
required facts about discrete valuations, forms of algebraic groups, and Galois
cohomology.
