The value for \Lambda, the cosmological constant, is tiny and positive. It's so tiny that as we take it to zero, the universe is still filled with clusters spiral and elliptical galaxies with a strong solid-angle-on-the-sky/brightness/redshift relation, and those galaxies filled with the same sorts of stars we see in the sky.

Indeed the evidence available prior to 1998 or so favoured a \Lambda of zero, and the early evidence favouring a tiny positive value was very much a surprise. The Hubble Space telescope had already been running and taking deep views for several years. COBE (https://science.nasa.gov/missions/cobe) and the Saskatoon experiment had already finished. None presaged the results of the Supernova Cosmology Project and High-Z Supernova Search Team. Follow-ons by Hubble (and others) and successors to COBE support the accelerated expansion.

The physical interpretation of the zero versus the tiny positive value is that in the former there was a last early acceleration which ended essentially all at once, with galaxy clusters then moving purely inertially; or alternatively there was an early acceleration which decayed, possibly in several steps, into a tiny persisting constant acceleration well before the formation of the surface of last scattering (the observed cosmic microwave background). Or alternatively there were mutiple sources of acceleration, one very large and which ceased early, and one which has been always-a-small constant.

There are different lines of evidence for how this residual acceleration has evolved. Practically all of it favours it settling down into one constant tiny value in the very early universe, and that the value can be determined with ever greater precision mostly by studying the fine detail in the cosmic microwave background and the various observables of highly red, dim, low-angular-diameter galaxies backlit by quasars and internally lit both by quasars and supernovae. The exact value is a matter of active research, as some of the data is conflicting.

None of the data disfavours some ongoing acceleration, and the conflict is generally within 10%. As an example of the physical consequences of the discrepency, the lower value means we can see more galaxies (and more galaxies will be able to see the light from the Crab supernova and other historical galactic supernovae), while the higher value fewer galaxies we can see and a lot fewer galaxies who could see a recent supernova within the Milky Way.

There are also various ideas that the tiny value of \Lambda is not constant but continues to evolve. Most of these unfailingly generate observables consistent with a new non-vanishing long-range force to accompany electromagnetism and gravitation, and such an extra ("fifth") force is almost wholly ruled out by evidence. Some store up this fifth force's energy until local matter-energy density is very low (trillions of trillions of years in our future) and unhide it then at various powers (often very high, in an essentially global phase change).

Attempts to remove ongoing acceleration altogether by setting \Lambda to zero seem somewhat contrived. A typical approach is to assume that the universe is much less homogeneous than it appears, and that we are being mislead by being in a highly unusual place exceptionally close to the centre of a large matter underdensity. This was of interest to several teams of theoreticians (Clifton, February) around 2008-2009, as they developed specific models -- within general relativity as the theory of gravitation -- in order to try to distinguish whole families of such models from the standard cosmology rather than outright advocating for those models in preference to the standard cosmology. More broadly, this is related to cosmologies that are wildly inhomogeneous at large scales compared to the standard cosmology, such that in some greater-than-galaxy-sized regions of spacetime the matter is much older than in others (in Wiltshire's voids, and in dense areas close to their boundaries, clocks run a lot more quickly so observers will see outside-the-void matter as gravitationally (collapse) rather than cosmologically (accelerated expansion) redshifted). Some of these models are designed to explore how one can generate an averaging procedure for general curved spacetimes, or at least for "lumpy" 3+1-dimensional ones, rather than to really challenge the most general form of the Copernican principle (i.e., they don't really want to put Earth pretty much at the centre of the universe).

Motivations do vary though; there are some productive everyday cosmologists whose single-author papers occasionally develop alternatives to the standard cosmology that fit closer to some scripture or other. There are also some modified gravity proponents (some of whom work every day with the standard cosmology) whose theories of gravitation "need help" from a favourable distribution of matter that picks out the region around us as atypical.