Studying connectivity of graphs and digraphs has various applications in many areas. Whenever different agents or entities perform an individual or combined task, which requires those agents to communicate or exchange information or goods in some way, it is of interest that this communication or exchange works as smooth and effective as possible. These kind of scenarios can usually be modelled as graphs or digraphs, where the vertices represent the agents or entities in question, and the edges or arcs represent the connections between them. Common examples include computer networks, communication networks, electricity grids, social networks, transport infrastructures, supply chains or other logistic scenarios, to name a few. In all these settings it is important to have a reliable connection between the agents, where the throughput through the network in question should be as high as possible. To achieve this goal it is convenient to study the underlying graphs and digraphs of these networks and provide some mathematical measurements, which give information about both the reliability and throughput of a network. In this thesis we deal with various parameters concerning the connectivity of graphs and digraphs, and present new results on these parameters themselves and how to maximize or optimize them in some sense. In particular, we take a closer look on local connectivity properties.