In 2013 scientists in Italy created a methodology for estimating the number of cells in various organs and tissues and thus the entire human body itself. There had been estimates previously, but with pretty large ranges.
The answer: a 5’7″ 154 pound human has 3.72×10^13 cells. That is 37.2 Trillion cells.
Link to their paper: http://www.tandfonline.com/doi/abs/10.3109/03014460.2013.807878