Membrane computing is a well-established and successful research field which belongs to the more general area of molecular computing. Membrane computing aims at defining parallel and non-deterministic computing models, called membrane systems or P Systems, which abstract from the functioning and structure of the cell. A membrane system consists of a spatial structure, a hierarchy of membranes which do not intersect, with a distinguishable membrane called skin surrounding all of them. A membrane without any other membranes inside is elementary, while a non-elementary membrane is a composite membrane. The membranes define demarcations between regions; for each membrane there is a unique associated region. Since we have a one-to-one correspondence, we sometimes use membrane instead of region, and vice-versa. The space outside the skin membrane is called the environment. In this thesis we define and investigate variants of systems of mobile membranes as models for molecular computing and as modelling paradigms for biological systems. On one hand, we follow the standard approach of research in membrane computing: defining a notion of computation for systems of mobile membranes, and investigating the computational power of such computing devices. Specifically, we address issues concerning the power of operations for modifying the membrane structure of a system of mobile membranes by mobility: endocytosis (moving a membrane inside a neighbouring membrane) and endocytosis (moving a membrane outside the membrane where it is placed). On the other hand, we relate systems of mobile membranes to process algebra (mobile ambients, timed mobile ambients, pi-calculus, brane calculus) by providing some encodings and adding some concepts inspired from process algebra in the framework of mobile membrane computing.