  A new family classification, based on a catalog of proper elements with $\sim
384,000$ numbered asteroids and on new methods is available. For the $45$
dynamical families with $>250$ members identified in this classification, we
present an attempt to obtain statistically significant ages: we succeeded in
computing ages for $37$ collisional families. We used a rigorous method,
including a least squares fit of the two sides of a V-shape plot in the proper
semimajor axis, inverse diameter plane to determine the corresponding slopes,
an advanced error model for the uncertainties of asteroid diameters, an
iterative outlier rejection scheme and quality control. The best available
Yarkovsky measurement was used to estimate a calibration of the Yarkovsky
effect for each family. The results are presented separately for the families
originated in fragmentation or cratering events, for the young, compact
families and for the truncated, one-sided families. For all the computed ages
the corresponding uncertainties are provided. We found 2 cases where two
separate dynamical families form together a single V-shape with compatible
slopes, thus indicating a single collisional event. We have also found 3
examples of dynamical families containing multiple collisional families, plus a
dubious case. We have found 2 cases of families containing a conspicuous
subfamily, such that it is possible to measure the slope of a distinct V-shape,
thus the age of the secondary collision. We also provide data on the central
gaps appearing in some families.
  The ages computed in this paper are obtained with a single and uniform
methodology, thus the ages of different families can be compared, providing a
first example of collisional chronology of the asteroid main belt.
