On the scale sizes of clusters of galaxies ~ 1 h-1 Mpc, many techniques have been applied to estimate m (e.g., Bahcall, this conference; Bahcall & Fan 1998; Dekel, Burstein & White 1997). The bottom line is that the apparent matter density appears to amount to only ~ 20-30% of the critical density required for a flat, = 1 universe. In fact, the most recent data are consistent with most of the extant data from the past 20 years. Cluster velocity dispersions (Carlberg et al. 1996), the distortion of background galaxies behind clusters or weak lensing (Kaiser & Squires 1993; Smail et al. 1995), the baryon density in clusters (White et al. 1993), and the existence of very massive clusters at high redshift (Bahcall & Fan), all currently favor a low value of the matter density (m ~ 0.2-0.3), at least on scales up to about 2h-1 Mpc.
However, measurements at scales larger than that of clusters are extremely challenging, and determinations of m have not yet converged (e.g., see Dekel, Burstein & White 1997). For example, measurements of peculiar velocities of galaxies, have led independent groups to come to very different conclusions, with estimates of m ranging from about 0.2 to 1.3. Dekel, Burstein & White conclude that the peculiar velocity results yield m > 0.3 at the 2- level. A new weak lensing study of a supercluster (Kaiser et al. 1999) on a scale of 6 h-1 Mpc, yields a (surprisingly) low value of m (~ 0.05), under the assumption that there is no bias in the way that mass traces light. Small & Sargent (1998) have recently probed the matter density for the Corona Borealis supercluster (at a scale of ~ 20 h-1 Mpc), finding m ~ 0.4. Under the assumption of a flat universe, global limits can also be placed on m from studies of type Ia supernovae (see next section); currently the supernova results favor a value m ~ 0.3.
The measurement of the total matter density of the Universe remains an important and challenging problem. It should be emphasized that all of the methods for measuring m are based on a number of underlying assumptions. For different methods, the list includes diverse assumptions about how the mass distribution traces the observed light distribution, whether clusters are representative of the Universe, the properties and effects of dust grains, or the evolution of the objects under study. The accuracy of any matter density estimate must ultimately be evaluated in the context of the validity of the underlying assumptions upon which the method is based. Hence, it is non-trivial to assign a quantitative uncertainty in many cases but, in fact, systematic effects (choices and assumptions) may be the dominant source of uncertainty.
An exciting result has emerged this year from atmospheric neutrino experiments undertaken at Superkamiokande (Totsuka, this volume), providing evidence for vacuum oscillations between muon and another neutrino species, and a lower limit to the mass in neutrinos. The contribution of neutrinos to the total density is likely to be small, although interestingly it may be comparable to that in stars.
Determining whether there is a significant, smooth underlying component to the matter density on the largest scales is a critical issue that must be definitively resolved. If, for example, some or all of the non-baryonic dark matter is composed of very weakly interacting particles, that component could prove very elusive and difficult to detect. It unfortunately remains the case that at present, it is not yet possible to distinguish unambiguously and definitively among m = 1, m + = 1, and open universes with 0 < 1, models all implying very different underlying fundamental physics. The preponderance of evidence at the present time, however, does not favor the simplest case of m = 1 (the Einstein-de Sitter universe).