  The zero temperature response of an interacting electron liquid to a
time-dependent vector potential of wave vector q and frequency w such that q <<
q_F, qv_F<< w << E_F/hbar (where q_F, v_F and E_F are the Fermi wavevector,
velocity and energy respectively) is equivalent to that of a continuous elastic
medium with nonvanishing shear modulus mu, bulk modulus K, and viscosity
coefficients eta and zeta. We establish the relationship between the
visco-elastic coefficients and the long-wavelength limit of the "dynamical
local field factors" G_{L(T)}(q,w), which are widely used to describe
exchange-correlation effects in electron liquids. We present several exact
results for mu, including its expression in terms of Landau parameters, and
practical approximate formulas for mu, eta and zeta as functions of density.
These are used to discuss the possibility of a transverse collective mode in
the electron liquid at sufficiently low density. Finally, we consider impurity
scattering and/or quasiparticle collisions at non zero temperature. Treating
these effects in the relaxation time (tau) approximation, explicit expressions
are derived for mu and eta as functions of frequency. These formulas exhibit a
crossover from the collisional regime (w tau << 1), where mu \sim 0 and eta
\sim n E_F tau, to the collisionless regime (w tau >> 1), where mu \sim nE_F
and eta \sim 0.
