Does anybody actually do astrometry in the 21st century? I mean the nitty-gritty measurements of positions of stars and planets using transit circles. The transit instruments I've seen are mostly in national observatories and are essentially museum pieces: in Greenwich, U.S. Naval Observatory in Washington, Ottawa, Sidney, Paris, etc. I've never read of anybody in the present time actually logging stars as they cross the meridian. If nobody is actually making these observations regularly, how do we know for sure the rotation period of the Earth and the exact positions of the planets? Modern astronomers study distant galaxies, but is anyone actually measuring the rotation of the Earth the old-fashioned way?
 A: Yes, there are astronometry instruments in continuous operation. 
For example, the Carlsberg Meridian Telescope at La Palma in the Canary Islands is used to provide accurate star positions to improve the proper motions of previously measured stars and for the measurement of solar system bodies, especially asteroids[1]. It[2]

... give[s] accurate positions of stars, allowing a reliable link to be 
  made between the bright stars measured by Hipparcos and the 
  fainter stars seen on photographic plates (as measured by 
  the APM and similar measuring machines). 
... the catalogues contain the improved proper motions of many 
  stars derived by combining the Carlsberg position with 
  previous epoch observations retrieved from a data-bank. The 
  average accuracy of the proper motions so derived is 0.003 
  arcsec/year. 

The Carlsberg Meridian Catalogue 14 was released in 2005 and is an astrometric and photometric catalogue of 96 million stars in the red magnitude range 9 to 17.
[1] Astronomical transit instrument (may or may not be hidden by a paywall) 
[2] The Carlsberg Meridian Telescope
A: These are not exactly 'transit' experiments but they are trying to achieve the same thing... There are currently two major astrometry missions in place at the moment, LSST is the Large Synoptic Sky Survey, This will map billions of objects and is essentially an update to the Hipparcos survey.
Secondly there is the GAIA mission which will involve a space based observation making a three dimensional map of the universe.  Gaia is taking great pains to make sure that each image can be tiled together accurately and then oriented on the sky.  This will give a very accurate representation of where we are...
A: I'm majoring in astrophysics, so my opinion may not be correct. Astrometry is dying, but will not die. It is applied in many fields of everyday life, which we may pay little attention to, as @Geoff Gaherty has mentioned. Moreover, although astromechanics, or celestial mechanics, is now able to predict astronomical phenomena in billion years, there are still so many factors to influence the results, so we need astrometry to do actual records. Besides, there is no effective method to measure the whole scale of the universe. If any, to measure the universe is still too huge a project to complete.
