With the current convergence of determinations of the Hubble Constant (e.g. The Extragalactic Distance Scale, 1997, Livio, Donahue and Panagia, eds.) to values within ±25% rather than a factor of two, and the clear possibility of determining q0 using high redshift supernovae (Garnavich et al. 1998), the major remaining problem in observational cosmology is the determination of Ω — what is the dark matter, how much is there, and how is it distributed?
The most direct approach to the last two parts of the question has been to study galaxy dynamics, first through the motions of galaxies in binaries, groups and clusters, and in the last decade and a half, driven by the observation of our motion w.r.t. the Cosmic Microwave Background (CMB) and thenotion that DM must be clumped on larger scales than galaxy clusters if (Ω is to be unity, through the study of large scale galaxy flows.
The ratio of the mass density to the closure mass density, Ω, is thought by most observers to be ~0.1-0.3, primarily based on the results of dynamical measurements of galaxy clusters and, more recently, gravitational lensing studies of clusters. In contrast, most theoretical cosmologists opt for a high density universe, Ω = 1.0, based on the precepts of the inflation scenario, the difficulty of forming galaxies in low density models given the observed smoothness of the microwave background radiation, and the observational evidence from the matching of the available large scale flow measurements (and the absolute microwave background dipole velocity) to the local density field. However this last result is extremely controversial—matching the velocity field to the density field derived from IRAS (60μ) selected galaxy samples yields high Ω values (e.g., Dekel et al. 1993) but matching to optically selected samples yields low values (Hudson 1994; Lahav et al. 1994; Santiago et al. 1995). On small scales, the high Ω camp argues that the true matter distribution is much more extended than the distribution of galaxies, so the dynamical mass estimates are biased low.