Introduction Tremendous progress in technology during the last 30 years has led to enormous improvements of accuracy in astrometry and related disciplines. Considering the growth of accuracy of positional observations in the course of time, we see that during the 25 years between 1988 and 2013 we expect the same gain in accuracy (4.5 orders of magnitude) as that realized during the whole previous history of astrometry, from Hipparchus till 1988 (over 2000 years). It is clear that for current and anticipated accuracy requirements, astronomical phenomena have to be formulated within the framework of general relativity. Many high-precision astronomical techniques already require sophisticated relativistic modeling since the main relativistic effects are several orders of magnitude larger than the technical accuracy of observations. Consequently, many current and planned observational projects cannot achieve their goals without a properly relativistic analysis. In principle, some basic relativistic effects have been taken into account since at least the 1960s. However, for a long time relativity has been viewed as “one more small correction” to the standard Newtonian formulae, rather than the fundamental framework for the data analysis. For the observational accuracy expected from proposed space instruments like Gaia, we must include not only the main relativistic effects, but also a multitude of second-order corrections. In this case it is impossible to treat relativistic effects as small corrections to the standard Newtonian scheme of data reduction. Consequently, the whole modeling scheme should be formulated in a language compatible with general relativity. Many Newtonian concepts and ideas should be re-considered and replaced by mathematically rigorous relativistic alternatives, including time, celestial coordinates, parallax, proper motion, etc.