  We discuss new models of an `affine' theory of gravity in multidimensional
space-times with symmetric connections. We use and develop ideas of Weyl,
Eddington, and Einstein, in particular, Einstein's proposal to specify the
space - time geometry by use of the Hamilton principle. More specifically, the
connection coefficients are determined using a `geometric' Lagrangian that is
an arbitrary function of the generalized (non-symmetric) Ricci curvature tensor
(and, possibly, of other fundamental tensors) expressed in terms of the
connection coefficients regarded as independent variables. Such a theory
supplements the standard Einstein gravity with dark energy (the cosmological
constant, in the first approximation), a neutral massive (or tachyonic) vector
field (vecton), and massive (or tachyonic) scalar fields. These fields couple
only to gravity and can generate dark matter and/or inflation. The new field
masses (real or imaginary) have a geometric origin and must appear in any
concrete model. The concrete choice of the geometric Lagrangian determines
further details of the theory, for example, the nature of the vector and scalar
fields that can describe massive particles, tachyons, or even `phantoms'. In
`natural' geometric theories, which are discussed here, dark energy must also
arise. We mainly focus on intricate relations between geometry and dynamics
while only very briefly considering approximate cosmological models inspired by
the geometric approach.
