This thesis is structured as follows. In Chapter 1, we provide background material about the concepts and techniques which are used in this thesis. In Chapter 2, we prove results which provide two new criteria for normal families of meromorphic functions, and which extend a recent result of Bergweiler and Langley. In Chapter 3, we extend a theorem of Bergweiler and Langley, and provide a result regarding the growth of a particular type of meromorphic function in an unbounded annulus. In Chapter 4, we extend two value distribution theorems of Langley and Zheng. In Chapter 5, we prove normal families and value distribution results in connection with composite functions.