In recent years, the science community has noticed a profound shift towards more interdisciplinarity as scientists with diverse backgrounds sought contact with each other and started new collaborations to bridge the gaps between existing research projects. A typical example hereof is the appearance of complexity science, an interdisciplinary research field par excellence which aims at devising accurate models to explain the dynamics of emergent collective phenomena in real-life interacting systems. The appearance of network structures in these models is rather common. The properties of the networks and the behaviour of the dynamical models are, for real-life events, often greatly influenced by the presence of a few elements with a very large influence: the so-called hubs, like for instance Google in the world wide web. Therefore, complexity science needs to make use of adjusted structures such as complex and scale-free networks. We introduce and discuss three dynamical models on complex and scale-free networks which illustrate how large-scale phenomena emerge as a consequence of the cooperative behaviour of the (microscopic) elements of the networks. A first topic deals with several percolation models ranging from a degree-dependent removal of links on scale-free networks to explosive processes in which large-scale structures appear very abruptly. Following on from this, a general network growth process with a fixed number of nodes and degree-dependent link addition probabilities is introduced. The percolation and network construction models substantiate how the details of microscopic link addition or removal processes influence both the large-scale characteristics of the constructed networks and the growth and disintegration of global network structures. In a last topic, the focus lies on the distribution of goods from producers to consumers and, in particular, we study the occurrence of breakdowns caused by a series of cascading failures in these distribution networks. We introduce an Ising-like spin model with quenched random fields that substantiates how these catastrophic black-outs, like power outages in the electricity grid, are caused by the solidarity between the individual suppliers.