Parsing Expression Grammars (PEGs) are a formalism that allow us to describe languages and that has as its distinguishing feature the use of an ordered choice operator. The class of languages described by PEGs properly contains all deterministic context-free languages. In this thesis we discuss the correspondence between PEGs and two other formalisms used to describe languages: regular expressions and Context-Free Grammars (CFGs). We present a new formalization of regular expressions that uses natural semantics and we show a transformation to convert a regular expression into a PEG that describes the same language; this transformation can be easily adapted to accommodate several extensions used by regular expression libraries (e.g., lazy repetition and independent subpatterns). We also present a new formalization of CFGs that uses natural semantics and we show the correspondence between right linear CFGs and equivalent PEGs. Moreover, we show that LL(1) grammars with a minor restriction define the same language when interpreted as a CFG and when interpreted as a PEG. Finally, we show how to transform strong-LL(k) CFGs into PEGs that are equivalent.