In this paper, as in other mathematical studies, a crystallographic group denotes a space group which is a discrete cocompact subgroup of the Euclidean group \(E(d)=\mathbb{R}^d\rtimes O(d)\). Space groups are not only important from the point of view of physics, as symmetry groups of crystals, but mathematically as well, because of interrelations between algebraic, arithmetic and metric properties expressible in terms of three celebrated theorems by Bieberbach and also occurring in other areas of mathematics. In particular, torsion-free space groups classify compact flat Riemannian manifolds. It is attractive, therefore, to extend some of the space group properties to algebraic groups, e. g. in the case of almost crystallographic groups, polycyclic groups and affine crystallographic groups. The new extension one finds in the paper, based on a generalization of Bieberbach's theorems, is called polycrystallographic group \(\Gamma\), for which a number of equivalent definitions are given. A first definition recasts that of a space group: \(\Gamma\) is a discrete cocompact subgroup of a semidirect product \(S\rtimes K\), with \(S\) a connected, simply connected solvable Lie group and \(K\) a compact subgroup of \(\Aut(S)\). Another one fits with Bieberbach's first theorem, replacing the previous \(K\) by a finite subgroup \(F\) of \(\Aut (S)\). Other definitions involve almost crystallographic groups and polycyclic groups, clarifying in this way their mutual relations. The analogue of Bieberbach's second theorem is based on the action of \(\Gamma\) on \(S\), metrically equivalent to that of \(\Gamma\) on a supersolvable Lie group determined by \(\Gamma\) up to an affine diffeomorphism. In the torsion-free case, \(\Gamma\) occurs as the fundamental group of a compact manifold, defined in terms of a torsion-free closed subgroup \(\Upsilon\subset S\rtimes K\). Finally, suitable embeddings allow a generalization of Bieberbach's third theorem.