  We review recent trends on inflationary dynamics in the context of viable
modified gravity theories. After providing a general overview of the
inflationary paradigm emphasizing on what problems of hot Big Bang theory
inflation solves, and a somewhat introductory presentation of single field
inflationary theories with minimal and non-minimal couplings, we review how
inflation can be realized in terms of several string motivated models of
inflation, which involve Gauss-Bonnet couplings of the scalar field, higher
order derivatives of the scalar field, and some subclasses of viable Horndeski
theories. We also present and analyze inflation in the context of Chern-Simons
theories of gravity, including various subcases and generalizations of string
corrected modified gravities which also contain Chern-Simons correction terms,
with the scalar field being identified with the invisible axion, which is the
most viable to date dark matter candidate. We also provide a detailed account
of vacuum $f(R)$ gravity inflation, and also inflation in $f(R,\phi)$ and
kinetic-corrected $f(R,\phi)$ theories of gravity. In the end of the review we
discuss the technique for calculating the overall effect of modified gravity on
the waveform of the standard general relativistic gravitational wave form.
