  In recent years the possibility of measuring the temporal change of radial
and transverse position of sources in the sky in real time have become
conceivable thanks to the thoroughly improved technique applied to new
astrometric and spectroscopic experiments, leading to the research domain we
call Real-time cosmology. We review for the first time great part of the work
done in this field, analysing both the theoretical framework and some endeavor
to foresee the observational strategies and their capability to constrain
models. We firstly focus on real time measurements of the overall redshift
drift and angular separation shift in distant source, able to trace background
cosmic expansion and large scale anisotropy, respectively. We then examine the
possibility of employing the same kind of observations to probe peculiar and
proper acceleration in clustered systems and therefore the gravitational
potential. The last two sections are devoted to the short time future change of
the cosmic microwave background, as well as to the temporal shift of the
temperature anisotropy power spectrum and maps. We conclude revisiting in this
context the effort made to forecast the power of upcoming experiments like
CODEX, GAIA and PLANCK in providing these new observational tools.
