Short answers, pulled from the story.
Cosmic inflation is the theory that the very early universe underwent exponential expansion, with distances between points doubling every 10 to the negative 37 seconds. Alan Guth first proposed it in 1979 while investigating why magnetic monopoles have never been observed. It was developed to resolve the flatness problem, the horizon problem, and the magnetic monopole problem that standard Big Bang cosmology could not explain.
The theory was developed through contributions from several physicists. Alexei Starobinsky at the Landau Institute for Theoretical Physics, Alan Guth at Cornell University, and Andrei Linde at Lebedev Physical Institute made the most notable early contributions. Guth coined the term "inflation" in January 1981. Starobinsky, Guth, and Linde won the 2014 Kavli Prize for pioneering the theory.
The COBE satellite in 1992 observed temperature anisotropies in the cosmic microwave background with a nearly scale-invariant spectrum, as inflation predicts. Data from the Planck spacecraft confirmed that the universe is flat to within a percent, homogeneous and isotropic to one part in 100,000, and that the spectral index n is 0.968 plus or minus 0.006, within the range of 0.92 to 0.98 that simple inflation models predict.
In many inflation models, quantum fluctuations can push the inflaton field upward in energy in some regions, causing those regions to inflate faster than they can stop. This means inflation never fully ends across the universe, producing a patchwork of infinitely many non-inflating regions, often called a multiverse. Paul Steinhardt, who introduced the first eternal inflation model in 1983, later became a critic of the inflationary paradigm partly because the multiverse prediction is considered untestable.
Roger Penrose argued from 1986 onward that inflation does not solve the initial conditions problem but simply displaces it. He calculated that obtaining a flat universe without inflation is more likely than obtaining one with inflation by a factor of ten to the googol power. At a 2015 conference he stated that inflation is not falsifiable and that it is falsified.
The inflaton is the hypothetical scalar field whose energy is thought to have driven the exponential expansion of the early universe. Its identity remains unknown. Alan Guth's early proposal that it might be the Higgs field is now considered problematic due to tension with electroweak-scale data. The potential energy curve of the inflaton can be shaped to fit a wide range of observations, which critics argue makes the theory difficult to falsify.