Einstein's theory of general relativity is without doubt one of greatest achievements in the history of Mankind. Even so, there are some ways, in which it leaves room for improvement. The last one hundred years and especially the last fifteen have seen many possibilities to remedy the small cracks in general relativity. Since the 1990s it has been known that the Universe is experiencing accelerating expansion. Explaining this with general relativity alone is not without problems. For this reason we need to and the viable alternatives to general relativity. 

While general relativity is based on certain assumptions, the various alternatives discard one or more of these assumptions for greater generality. One path leads to f(R) theories of gravity, which let the gravitational action be a function of the Ricci curvature scalar instead of the plain linear term in general relativity. Thus, there is an infinite number of possible f(R) gravity models. 

Many of these possible f(R) models can be ruled out as unphysical from the start. However, it is possible to construct models, which seem to fit observations even better than the highly successful general relativity with the cosmological constant. Even for these models, there might still be lurking some dynamics or other characteristics, which render them unphysical. Further constraining the class of viable f(R) theories provides us with better understanding of gravity itself and the characteristics required of a new gravitational theory. As such it paves way for understanding the needs of a working quantum gravity theory. 

In this thesis I develop methods to better constrain viable f(R) models and apply these methods to select models. I use both theoretical tools to examine the mathematical background of f(R) for instabilities and link results to observational data. Even a mathematically sound candidate for a physical theory must stand trial to observations. 

The methods I develop in this thesis can be applied to a wide range of f(R) models for tests of viability. As the body of available data grows and the observations become ever more precise, these methods will provide even more stringent bounds and rule out more models. Many of the methods can also be used other modified gravity theories besides f(R) gravity.