  We were asked to debate the value of the cosmological mass-density parameter
Omega. Is Omega_m=1 in accordance with the simplest model? Is Omega_m much
smaller as indicated by some observations? There is conflicting evidence. We
lay out the various methods for measuring Omega_m, mention new developments and
current estimates, and focus on prospects versus the associated difficulties.
We try to shed light on the uncertainties that are responsible for the span of
estimates for Omega_m.
  We divide the methods into the following four classes:
  a. Global measures of the properties of space-time that constrain
combinations of Omega_m and the other cosmological parameters (Lambda, Ho, to).
  b. Virialized Systems. Nonlinear dynamics within galaxies and clusters on
comoving scales 1-10 Mpc/h.
  c. Large-scale structure. Mildly-nonlinear gravitational dynamics of
fluctuations on scales 10-100 Mpc/h, in particular cosmic flows.
  d. Growth rate of fluctuations from the last scattering of the CMB or from
high redshift galaxies to the present.
  Methods (b) typically yield low values of Omega_m=0.2-0.3. Methods (a) and
(c) typically indicate higher values of Omega_m=0.4-1.
