  We investigate aging continuous time random walks (ACTRW), introduced by
Monthus and Bouchaud [{\em J. Phys. A} {\bf 29}, 3847 (1996)]. Statistical
behaviors of the displacement of the random walker ${\bf r}={\bf r}(t) - {\bf
r}(0)$ in the time interval $(0,t)$ are obtained, after aging the random walk
in the time interval $(-t_a,0)$. In ACTRW formalism, the Green function $P({\bf
r}, t_a, t)$ depends on the age of the random walk $t_a$ and the forward time
$t$. We derive a generalized Montroll--Weiss equation, which yields an exact
expression for the Fourier double Laplace transform of the ACTRW Green
function. Asymptotic long times $t_a$ and $t$ behaviors of the Green function
are investigated in detail. In the limit of $t\gg t_a$, we recover the standard
non-equilibrium CTRW behaviors, while the important regimes $t\ll t_a$ and $t
\simeq t_a$ exhibit interesting aging effects. Convergence of the ACTRW results
towards CTRW behavior, becomes extremely slow when the diffusion exponent
becomes small. In the context of biased ACTRW, we investigate an aging Einstein
relation. We briefly discuss aging in Scher-Montroll type of transport in
disordered materials.
