  We perform a systematic search for N=1 Minkowski vacua of type II string
theories on compact six-dimensional parallelizable nil- and solvmanifolds
(quotients of six-dimensional nilpotent and solvable groups, respectively).
Some of these manifolds have appeared in the construction of string backgrounds
and are typically called twisted tori. We look for vacua directly in ten
dimensions, using the a reformulation of the supersymmetry condition in the
framework of generalized complex geometry. Certain algebraic criteria to
establish compactness of the manifolds involved are also needed. Although the
conditions for preserved N=1 supersymmetry fit nicely in the framework of
generalized complex geometry, they are notoriously hard to solve when coupled
to the Bianchi identities. We find solutions in a large-volume,
constant-dilaton limit. Among these, we identify those that are T-dual to
backgrounds of IIB on a conformal T^6 with self-dual three-form flux, and hence
conceptually not new. For all backgrounds of this type fully localized
solutions can be obtained. The other new solutions need multiple intersecting
sources (either orientifold planes or combinations of O-planes and D-branes) to
satisfy the Bianchi identities; the full list of such new solution is given.
These are so far only smeared solutions, and their localization is yet unknown.
Although valid in a large-volume limit, they are the first examples of
Minkowski vacua in supergravity which are not connected by any duality to a
Calabi-Yau. Finally, we discuss a class of flat solvmanifolds that may lead to
AdS_4 vacua of type IIA strings.
