  We use cosmography to present constraints on the kinematics of the Universe,
without postulating any underlying theoretical model. To this end, we use a
Monte Carlo Markov Chain analysis to perform comparisons to the supernova Ia
Union 2 compilation, combined with the Hubble Space Telescope measurements of
the Hubble constant, and the Hubble parameter datasets. We introduce a sixth
order cosmographic parameter and show that it does not enlarge considerably the
posterior distribution when comparing to the fifth order results. We also
propose a way to construct viable parameter variables to be used as
alternatives of the redshift $z$. These can overcome both the problems of
divergence and lack of accuracy associated with the use of $z$. Moreover, we
show that it is possible to improve the numerical fits by re-parameterizing the
cosmological distances. In addition, we constrain the equation of state of the
Universe as a whole by the use of cosmography. Thus, we derive expressions
which can be directly used to fit the equation of state and the pressure
derivatives up to fourth order. To this end, it is necessary to depart from a
pure cosmographic analysis and to assume the Friedmann equations as valid. All
our results are consistent with the $\Lambda$CDM model, although alternative
fluid models, with nearly constant pressure and no cosmological constant, match
the results accurately as well.
