Inflation is the name for several models of the very evolution of the universe. It's sort of an "add-on" to the Big Bang Model. The main problems with the early universe are:
(a) It is very flat (b) There are very little gravitational waves (c) The universe looks roughly the same in each direction (d) There are no magnetic monopoles (e) Galaxies look like they formed from fluctuations in the homogenous soup of the early universe.
Both (a) (b) (c) are quite non-generic. The universe from current understanding could have been more curved, look different in different directions and should have plenty of gravitational radiation.
Inflation attempts to solves all of these. However it does quite poorly at (b), for (c) it doesn't seem to match precisely how the galaxies formed and (d) seems like less of an issue today as it was a problem suggested by theories popular in the 80s.
Certainly some of the extreme models of inflation get into very ambiguous territory where it is hard to know if you could every disprove them because the parameters can be adjusted to stay consistent with observations.
For the decent models of inflation they seem to explain some things but not all.
It's very hard to know how the science of the early universe is going to continue since we've learned a lot about quantum mechanics in the last decade that suggests parts of the world are not amenable to mathematical description and this is probably very relevant in the early universe. Probably plenty of work ahead of us.
Edited by Son Goku, : Don't write posts on the phone when you're in a rush