  We review inflationary cosmology in modified gravity such as $R^2$ gravity
with its extensions in order to generalize the Starobinsky inflation model. In
particular, we explore inflation realized by three kinds of effects:
modification of gravity, the quantum anomaly, and the $R^2$ term in loop
quantum cosmology. It is explicitly demonstrated that in these inflationary
models, the spectral index of scalar modes of the density perturbations and the
tensor-to-scalar ratio can be consistent with the Planck results. Bounce
cosmology in $F(R)$ gravity is also explained.
