  We introduce and develop the 1+3 covariant approach to relativity and
cosmology to spacetimes of arbitrary dimensions that have nonzero torsion and
do not satisfy the metricity condition. Focusing on timelike observers, we
identify and discuss the main differences between their kinematics and those of
their counterparts living in standard Riemannian spacetimes. At the centre of
our analysis lies the Raychaudhuri equation, which is the fundamental formula
monitoring the convergence/divergence, namely the collapse/expansion, of
timelike worldline congruences. To the best of our knowledge, we provide the
most general expression so far of the Raychaudhuri equation, with applications
to an extensive range of non-standard astrophysical and cosmological studies.
Assuming that metricity holds, but allowing for nonzero torsion, we recover the
results of analogous previous treatments. Focusing on non-metricity alone, we
identify a host of effects that depend on the nature of the timelike congruence
and on the type of the adopted non-metricity. We also demonstrate that in
spaces of high symmetry one can recover the pure-torsion results from their
pure non-metricity analogues, and vice-versa, via a simple ansatz between
torsion and non-metricity.
