We show that the tails of the single-point velocity probability distri
bution function (PDF) are generally non-Gaussian in developed turbulen
ce. By using instanton formalism for the randomly forced Navier-Stokes
equation, we establish the relation between the PDF tails of the velo
city and those of the external forcing. In particular, we show that a
Gaussian random force having correlation scale L and correlation time
tau produces velocity PDF tails In P(v) proportional to - v(4) at v mu
ch greater than v(rms), L/tau. For a short-correlated forcing when tau
much less than L/v(rms) there is an intermediate asymptotics In P(v)
proportional to - v(3) at L/tau much greater than v much greater than
v(rms).