  In Chapter 1 I present the current picture of the universe and briefly review
the cosmological constant problem and some of the theories proposed to solve
it. The following Chapters essentially contain the published papers with some
modifications and additions: in Chapter 2 I study the radiation-dominated epoch
in $R^2$ gravity, gravitational particle production and its implications for
the early and contemporary universe; in Chapter 3 I investigate the formation
of curvature singularities in contracting astronomical systems; in Chapter 4, I
discuss some mechanisms to prevent such singularities and calculate the related
cosmic-ray production; finally, in Chapter 5, I deal with spherically symmetric
solutions and explore the possibility of gravitational repulsion.
