  We study the aging property for stationary models in the KPZ universality
class. In particular, we show aging for the stationary KPZ fixed point, the
Cole-Hopf solution to the stationary KPZ equation, the height function of the
stationary TASEP, last-passage percolation with boundary conditions and
stationary directed polymers in the intermediate disorder regime. All of these
models are shown to display a universal aging behavior characterized by the
rate of decay of their correlations. As a comparison, we show aging for models
in the Edwards-Wilkinson universality class where a different decay exponent is
obtained. A key ingredient to our proofs is a characteristic of space-time
stationarity - covariance-to-variance reduction - which allows to deduce the
asymptotic behavior of the correlations of two space-time points by the one of
the variances at one point. We formulate several open problems.
