Discrepancies between observations at early and late cosmic epochs, and the
vacuum energy problem associated with the interpretation of cosmological
constant, are questioning the $\Lambda$CDM model. Motivated by these conceptual
and observational facts, extensions of Einstein’s gravity are recently
intensively considered in view of curing unsolved issues suffered by General
Relativity at ultraviolet and infrared scales. Here, we provide a short
overview of some aspects of $f(R)$ gravity, focusing, in particular, on
cosmological applications. Specifically, Noether symmetries are adopted as a
criterion to select viable models and investigate the corresponding dynamics.
We thus find solutions to the cosmological field equations, analyzing the
behaviour of selected models from the matter-dominated to the present epoch.
Moreover, constraints coming from energy conditions and the so-called swampland
criteria are also considered. In particular, we qualitatively discuss the
possibility of $f(R)$ gravity to account for fixing cosmic tensions.
