  We generalize Nozi\`eres' Fermi-liquid theory for the low-energy behavior of
the Kondo model to that of the single-impurity Anderson model. In addition to
the electrons' phase shift at the Fermi energy, the low-energy Fermi-liquid
theory is characterized by four Fermi-liquid parameters: the two given by
Nozi\`eres that enter to first order in the excitation energy, and two
additional ones that enter to second order and are needed away from
particle-hole symmetry. We express all four parameters in terms of
zero-temperature physical observables, namely the local charge and spin
susceptibilities and their derivatives with respect to the local level
position. We determine these in terms of the bare parameters of the Anderson
model using Bethe Ansatz and Numerical Renormalization Group (NRG)
calculations. Our low-energy Fermi-liquid theory applies throughout the
crossover from the strong-coupling Kondo regime via the mixed-valence regime to
the empty-orbital regime. From the Fermi-liquid theory, we determine the
conductance through a quantum dot symmetrically coupled to two leads in the
regime of small magnetic field, low temperature and small bias voltage, and
compute the coefficients of the $\sim B^2$, $\sim T^2$, and $\sim V^2$ terms
\textit{exactly} in terms of the Fermi-liquid parameters. The coefficients of
$T^2$, $V^2$ and $B^2$ are found to change sign during the Kondo to
empty-orbital crossover. The crossover becomes universal in the limit that the
local interaction is much larger than the level width. For completeness, we
also compute the shot noise and discuss the resulting Fano factor.
