  Spacetime geometry is described by two -- {\em a priori} independent --
geometric structures: the symmetric connection $\Gamma$ and the metric tensor
$g$. Metricity condition of $\Gamma$ (i.e. $\nabla g = 0$) is implied by the
Palatini variational principle, but only when the matter fields belong to an
exceptional class. In case of a generic matter field, Palatini implies
non-metricity of $\Gamma$. Traditionally, instead of the (1st order) Palatini
principle, we use in this case the (2nd order) Hilbert principle, assuming
metricity condition {\em a priori}. Unfortunately, the resulting right-hand
side of the Einstein equations does not coincide with the matter
energy-momentum tensor. We propose to treat seriously the Palatini-implied
non-metric connection. The conventional Einstein's theory, rewritten in terms
of this object, acquires a much simpler and universal structure. This approach
opens a room for the description of the large scale effects in General
Relativity (dark matter?, dark energy?), without resorting to purely
phenomenological terms in the Lagrangian of gravitational field. All theories
discussed in this paper belong to the standard General Relativity Theory, the
only non-standard element being their (much simpler) mathematical formulation.
As a mathematical bonus, we propose a new formalism in the calculus of
variations, because in case of hyperbolic field theories the standard approach
leads to nonsense conclusions.
