By observing the velocity at which galaxies are spreading out in the cosmos, and the average density of visible matter (and therefore the slowing effect of gravity on the spread of galaxies) we can calculate how long it has been since these galaxies were all crammed together. This gives us a good first estimate of the age of the universe.
However, it appears from observation of space from long ago ( we can see what space looked like a long time in the past by observing deep space; due to the finite speed of light, long ago, distant events can be observed here today) it appears that the universe went through at least one "inflationary" phase, during which gravity seems to have accelerated the galaxies away from each other, rather than slow their spread.
Answered by:
Rob Landolfi, Science Teacher, Washington, DC

It must be the case that the oldest objects in the universe are younger than the universe itself. So looking at globular clusters for example would give a lower band on the age.
As all know, the universe is expanding. A critical parameter is the scale factor a(t), this tells us about the expansion rate of the universe.
Knowing a(t) would therefore make it possible to predict the age of the universe.
At a rough estimate, the rate of expansion is given by the Hubble parameter H = (da/dt)/a. This means our first guess at the age of the universe should be the time scale associated with the Hubble parameter, H^{-1}. This gives
H^-1 = 9.77 h^{-1} x 10^{9} Years.
(0.55 < h < 0.75 which measures the uncertainty in H)
This is known as Hubble time.
Better theoretical estimates can be found by solving the Friedmann equation for a(t). The Friedmann equation tells us how a(t) evolves taking into account what kind of matter is in the universe and the geometry on the universe.
Another estimate (based on the Friedmann equation) would be to use a dust dominated universe, no cosmological constant and not worry about the geometry. This produces a factor of 2/3 the time estimate. So the universe is younger than out first estimate.
The only way to really pin down the age is through observational cosmology and trying to get accurate values for the expansion parameter and Hubble's parameter. Another spanner in the works is the possibility of a non-zero cosmological constant, indeed some recent observations suggest that we are now in a phase of inflation.
But as a rough figure, the age of the universe is ~ 10^{9} years. This could be revised at any time...
Answered by:
Andrew James Bruce, Grad Student UK

'There comes a time when the mind takes a higher plane of knowledge but can never prove how it got there. All great discoveries have involved such a leap. The important thing is not to stop questioning.'