We review recent trends in inflationary dynamics in the context of viable modified gravity theories. After providing a general overview of the inflationary paradigm emphasizing on what problems 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 $f\left(R\right)$ gravity inflation, and also inflation in $f(R,\varphi )$ and kinetic-corrected $f(R,\varphi )$ theories of gravity. At 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.

We study how a cosmological bounce, with a type IV singularity at the bouncing point, can be generated by a classical vacuum $F(G)$ gravity. We focus our investigation on the behavior of the vacuum $F(G)$ theory near the type IV singular bouncing point and address the stability of the resulting solution by treating the equations of motion as a dynamical system. In addition, we investigate how the scalar perturbations of the background metric evolve, emphasizing cosmological times near the type IV singular bouncing point. Finally, we also investigate which mimetic vacuum $F(G)$ gravity can describe the singular bounce cosmology.