In 1988 Shafarevich asked me to write a volume for the Encyclopaedia of
Mathematical Sciences on Diophantine Geometry. I said yes, and here is
the volume. By definition, diophantine problems concern the solutions of
equations in integers, or rational numbers, or various generalizations,
such as finitely generated rings over Z or finitely generated fields
over Q. The word Geometry is tacked on to suggest geometric methods.
This means that the present volume is not elementary. For a survey of
some basic problems with a much more elementary approach, see [La
9Oc]. The field of diophantine geometry is now moving quite rapidly.
Out- standing conjectures ranging from decades back are being proved. I
have tried to give the book some sort of coherence and permanence by em-
phasizing structural conjectures as much as results, so that one has a
clear picture of the field. On the whole, I omit proofs, according to
the boundary conditions of the encyclopedia. On some occasions I do give
some ideas for the proofs when these are especially important. In any
case, a lengthy bibliography refers to papers and books where proofs may
be found. I have also followed Shafarevich's suggestion to give
examples, and I have especially chosen these examples which show how
some classical problems do or do not get solved by contemporary in-
sights. Fermat's last theorem occupies an intermediate position. Al-
though it is not proved, it is not an isolated problem any more.
