Linear dynamics is an area of mathematics, lying in the intersection of operator theory and dynamical systems, and consisting in the study of iter- ates of linear operators. It has received a lot of attention in the last decades, producing a flurry of intriguing results. In this thesis, within a global view on the subject, new results are presented, obtained during the three years as PhD student in Mathematical Analysis, at the Department of Mathematics and Physics in Universit`a degli Studi della Campania "L. Vanvitelli". The thesis consists of five chapters. Starting, in Chapter 1, with a brief introduction to the modern theory of linear dynamical systems, it provides an exploration of the fundamental and fascinating notions of linear dynam- ics, with particular emphasis on hypercyclicity, topological mixing, Devaney and Li-Yorke chaos, frequent hypercyclicity, generalized hyperbolicity, ex- pansivity and shadowing. These properties are completely characterized for a significant class of operators, the weighted shifts, to the presentation of which part of Chapter 2 is dedicated, and frequently recurring through the thesis. Then, in the second part of Chapter 2 and in Chapter 3, the study moves toward the investigation of the above mentioned notions in the con- text of a versatile and meaningful, under the dynamical point of view, class of operators: the composition operators, i.e., Tf : ? ? ? ? f, for which it is the nature of the transformation f to determine the dynamics. Among other results in the thesis, new characterizations of generalized hyperbolic- ity, shadowing and expansivity are presented and proved in the context of composition operators, as well as, in Chapter 4, versatile techniques which allow to lift up characterizations given for weighted shifts to a broader class of composition operators, called shift-like operators. Finally, although the thesis is mainly centred on the linear setting, it closes, in Chapter 5, with an overview on properties of non-linear composition operators also, thus pro- viding a comparison between the two types of composition operators and showing that the theory does not seem to run in parallel for the two cases.