  This review article aims at presenting the theory of inflation. We first
describe the background spacetime behavior during the slow-roll phase and
analyze how inflation ends and the Universe reheats. Then, we present the
theory of cosmological perturbations with special emphasis on their behavior
during inflation. In particular, we discuss the quantum-mechanical nature of
the fluctuations and show how the uncertainty principle fixes the amplitude of
the perturbations. In a next step, we calculate the inflationary power spectra
in the slow-roll approximation and compare these theoretical predictions to the
recent high accuracy measurements of the Cosmic Microwave Background radiation
(CMBR) anisotropy. We show how these data already constrain the underlying
inflationary high energy physics. Finally, we conclude with some speculations
about the trans-Planckian problem, arguing that this issue could allow us to
open a window on physical phenomena which have never been probed so far.
