In Chapter 1 we introduce many of the concepts and techniques, including Nevanlinna theory, referred to throughout the rest of the thesis. In Chapter 2 we extend a result of Langley and Shea concerning the distribution of zeros of the logarithmic derivative of meromorphic functions to higher order logarithmic derivatives. Chapter 3 details an alternative formulation, avoiding reference to the multiplicity of poles, of a result due to Chuang concerning differential polynomials. In Chapter 4 we generalise a theorem of Bergweiler and Eremenko concerning transcendental singularities of the the inverse of a meromorphic function. In Chapter 5 we generalise a result of Gordon to show that an unbounded analytic function on a quasidisk has a strong form of unboundedness there. Chapter 6 contains a proof of a result concerning the normality of families of analytic functions such that the composition of any of these functions with a fixed (meromorphic) outer factor has no fixpoints in a given domain.

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.

L'objectif de cette thèse, est l'étude de la convergence d'applications méromorphes entre deux variétés U et X. D'abord nous rappelons trois types de convergence d'applications méromorphes: Convergence forte, Convergence faible et Gamma-convergence. Notre premier résultat est que la convergence forte est équivalente à la convergence au sens de cycles. Une caractéristique agréable de la convergence faible et la convergence Gamma est que les ensembles de convergence sont localement pseudoconvexes à condition que la variété X soit de Gauduchon. Notre deuxième résultat est dans le cas d'applications méromorphes à valeurs dans l'espace projectif complexe. Nous montrons que la convergence Gamma est équivalente à la convergence au sens de Fujimoto. La convergence faible est équivalente à la convergence Gamma à condition que la représentation de l'application limite soit aussi irréductible. La convergence forte est équivalente à la convergence faible à condition que que les masses Monge-Ampère non-pluripolaires convergent. Un exemple de A. Rashkovskii montre que les volumes des graphes d'une suite d'applications méromorphes qui converge faiblement peuvent augmenter sur tout compact de la variété source U, dans le cas ou la dimension de deux variétés est strictement supérieur à 2. Finalement, nous prouvons le résultat suivant: Si une famille d'applications méromorphes, du bidisque dans une surface complexe compacte, est équicontinue dans un voisinage de la frontière, alors le volume des graphes est localement uniformément borné. / This thesis is concerned with study of convergence of meromorphic mappings between complex manifolds. First we racall three types of convergence of meromorphic mappings: strong convergence, weak convergence and Gamma convergence. Our first result is that the strong convergence is equivalent to the convergence of graphs in the topology of cycles. A nice feature of weak and Gamma convergence is that the set of convergence is locally pseudoconvex provided that the manifold X is Kahler, or even Gauduchon. Our second result concerne the convergence of meromorphic mappings with values in complex projective space. We show that Gamma convergence is equivalent to convergence in the sense of Fujimoto. Weak convergence is equivalent to the Gamma plus the representation of limit maps should be reduced. Strong convergence is equivalent to weak convergence plus the corresponding non-pluripolar Monge-Ampere masses should converge. An example of A. Raskovskii shows that the volumes of graphs of a weakly converging sequence of meromorphic mappins in the case when the dimension greater than 2, unlike to the case of strong convergence may grow over compacts in the source manifold. Finally, we prove that a family of meromorphic mappings from a bidisc to a compact complex surface, which are equicontinuous in a neighborhood of the boundary of the bidisc, has the volumes of its graphs locally uniformly bounded.