Currently (Sept 2014), the number of digits is 10 trillion (and fifty).
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Pi was calculated to 10 trillion digits on October 17, 2011.
As of April 1999, 68.7 billion places had been calculated. As of September 1999, 206 billion places had been calculated.
1.2411 trillion digits (1,241,100,000,000) digits of pi have been dicovered. * * * * * 1.2 trillion? That is ancient history! Pi is a transcendental number and so its decimal representation has infinitely many digits. Ten trillion digits in its decimal form were calculated in October 2011. See link for details.
The "Pi Computer Project" or PCP finished some time in November 2010, it calculated about 5 trillion digits! * * * * * But the number crunching goes on! As of 17 October 2011, the answer is 10 trillion.
There are infinitely many. In October 2011, Alexander J Yee and Shigeru Kondo has calculated 10 trillion (and 50) digits. That is the record as at the time of writing this answer (June 2013).