The simplest models of inflation, those in which only one scalar field is present and is minimally coupled, lead to very simple and clear predictions: (1) the Universe is simply connected, (nearly) homogeneous and isotropic, with no detectable large-scale rotation, at least up to scales slightly larger than the present particle horizon; (2) the observable Universe is spatially flat; (3) the scalar perturbations that eventually originated the large-scale structures observed today, if generated during inflation, were primordial (i.e. passive), adiabatic and Gaussian distributed, with a nearly scale-invariant power spectrum.
However, in more complicated models, where more than one scalar field is present, with the possibility of the such fields being strongly coupled, most of the above predictions can be weakened. The exceptions are those of a trivial topology, the absence of large-scale rotation, and the primordial nature of the scalar perturbations. Nevertheless, these are still sufficient to provide tests for both hypothesis: of inflation as the event responsible for the present-day large-scale Universe being nearly homogeneous, isotropic and spatially flat; and of inflation as the most important mechanism behind the generation of the scalar perturbations that eventually originated the large-scale structures observed in the Universe today.
I will focus on the later hypothesis [see (55) for a general review of inflation], describing the observational tests, and present constraints, on the various characteristics expected for scalar perturbations in the simplest inflationary models. These characteristics have an impact both on the nature and timescale for the formation and evolution of structures on large-scale in the Universe, and on the properties of the cosmic microwave background radiation (CMBR). I will therefore spend most of this review on how the present observational data regarding these two topics constrain the characteristics of the scalar perturbations, and what in turn that tells us about inflation. I will also discuss the prospects of detecting locally a possible stochastic background of gravitational waves produced during inflation, and the present evidence regarding the geometry of the Universe, and whether it supports the prediction of spatial flatness associated with the simplest inflationary models. Though tensor perturbations and spatial flatness cannot be used to test inflation, given that inflationary models exist which do not predict them, they can offer strong support to the hypothesis that an inflationary period did occur in the early Universe.