  For chaotic cavities with scattering leads attached, transport properties can
be approximated in terms of the classical trajectories which enter and exit the
system. With a semiclassical treatment involving fine correlations between such
trajectories we develop a diagrammatic technique to calculate the moments of
various transport quantities. Namely, we find the moments of the transmission
and reflection eigenvalues for systems with and without time reversal symmetry.
We also derive related quantities involving an energy dependence: the moments
of the Wigner delay times and the density of states of chaotic Andreev
billiards, where we find that the gap in the density persists when subleading
corrections are included. Finally, we show how to adapt our techniques to
non-linear statistics by calculating the correlation between transport moments.
In each setting, the answer for the $n$-th moment is obtained for arbitrary $n$
(in the form of a moment generating function) and for up to the three leading
orders in terms of the inverse channel number. Our results suggest patterns
which should hold for further corrections and by matching with the low order
moments available from random matrix theory we derive likely higher order
generating functions.
