Arakelov theory is a new geometric approach to diophantine equations. It
combines algebraic geometry, in the sense of Grothendieck, with refined
analytic tools such as currents on complex manifolds and the spectrum of
Laplace operators. It has been used by Faltings and Vojta in their
proofs of outstanding conjectures in diophantine geometry. This account
presents the work of Gillet and Soulé, extending Arakelov geometry to
higher dimensions. It includes a proof of Serre's conjecture on
intersection multiplicities and an arithmetic Riemann-Roch theorem. To
aid number theorists, background material on differential geometry is
described, but techniques from algebra and analysis are covered as well.
Several open problems and research themes are also mentioned.