  We investigate the cosmic age problem associated with the old high-$z$ quasar
APM 08279 + 5255 and the oldest globular cluster M 107, both being difficult to
accommodate in $\Lambda$CDM model. By evaluating the age of the Universe in a
model that has an extremely phantom like form of dark energy (DE), we show that
simply introducing the dark energy alone does not remove the problem, and the
interaction between dark matter (DM) and DE need to be taken into account.
Next, as examples, we consider two interacting DE models. It is found that both
these two interacting DE Models can predict a cosmic age much greater than that
of $\Lambda$CDM model at any redshift, and thus substantially alleviate the
cosmic age problem. Therefore, the interaction between DM and DE is the crucial
factor required to make the predicted cosmic ages consistent with observations.
