I will present selected field theoretical aspects and Bayesian model selection studies in a particular class of modified gravity theories, so-called nonlocal gravity theories. In particular, I will focus on three nonlocal gravity models that have been proposed for explaining the late-time acceleration of the expansion of the universe and have been shown to provide a statistically equivalent fit to LCDM given recent cosmological data. I will review their phenomenology relevant for different cosmological probes such as the cosmic microwave background, supernovae, large scale structure, as well as future gravitational waves interferometers, and also discuss the issue of cosmological screening mechanism in these models.