  Recent developments in ergodic theory, additive combinatorics, higher order
Fourier analysis and number theory give a central role to a class of algebraic
structures called nilmanifolds. In the present paper we continue a program
started by Host and Kra. We introduce nilspaces as structures satisfying a
variant of the Host-Kra axiom system for parallelepiped structures. We give a
detailed structural analysis of abstract and compact topological nilspaces.
Among various results it will be proved that compact nilspaces are inverse
limits of finite dimensional ones. Then we show that finite dimensional compact
connected nilspaces are nilmanifolds. The theory of compact nilspaces is a
generalization of the theory of compact abelian groups. This paper is the main
algebraic tool in the second authors approach to Gowers's uniformity norms and
higher order Fourier analysis.
