A general approach to viable modified $f(R)$ gravity is developed in both the Jordan and the Einstein frames. A class of exponential, realistic modified gravities is introduced and investigated with care. Special focus is made on step-class models, most promising from the phenomenological viewpoint and which provide a natural way to classify all viable modified gravities. One- and two-step models are explicitly considered, but the analysis is extensible to $N$-step models. Both inflation in the early universe and the onset of recent accelerated expansion arise in these models in a natural, unified way. Moreover, it is demonstrated that models in this category easily pass all local tests, including stability of spherical body solution, nonviolation of Newton’s law, and generation of a very heavy positive mass for the additional scalar degree of freedom.

Dark energy cosmology is considered in a modified Gauss-Bonnet (GB) model of gravity where an arbitrary function of the GB invariant, $f(G)$, is added to the general relativity action. We show that a theory of this kind is endowed with a quite rich cosmological structure: it may naturally lead to an effective cosmological constant, quintessence, or phantom cosmic acceleration, with a possibility for the transition from deceleration to acceleration. It is demonstrated in the paper that this theory is perfectly viable, since it is compliant with the solar system constraints. Specific properties of $f(G)$ gravity in a de Sitter (dS) universe, such as dS and SdS solutions, their entropy, and its explicit one-loop quantization are studied. The issue of a possible solution of the hierarchy problem in modified gravities is also addressed.

We consider scalar-Gauss-Bonnet and modified Gauss-Bonnet gravities and reconstruct these theories from the universe expansion history. In particular, we are able to construct versions of those theories (with and without ordinary matter), in which the matter-dominated era makes a transition to the cosmic acceleration epoch. In several of the cases under consideration, matter dominance and the deceleration-acceleration transition occur in the presence of matter only. The late-time acceleration epoch is described asymptotically by de Sitter space but may also correspond to an exact $\Lambda \mathrm{CDM}$ cosmology, having in both cases an effective equation of state parameter $w$ close to $-1$. The one-loop effective action of modified Gauss-Bonnet gravity on the de Sitter background is evaluated and it is used to derive stability criteria for the ensuing de Sitter universe.