What do the bridges of Königsberg, synaptically connected neurons in our brains and the galaxies illuminating the dark voids of our universe have in common? All of these real-world phenomena can be described as collections of discrete discernible objects which are interlinked to form weblike structures called networks. This chapter will introduce the mathematical representation of such networks, and familiarise us with the basic concepts, ideas and terminology of a vast and ever-growing research field whose roots date back to the work of Leonard Euler. By taking a closer look at a number of concrete network models - specifically the random graph models which prominently feature as descriptive vessels for many natural phenomena - and briefly exploring some deep-rooted conceptual limitations of these models, we hope not only to motivate the need for a rigorous mathematical framework for the study of networks at finite scales, but also to accentuate the potential advantages of a more dynamical vantage point from which to view networks and their defining characteristics in later chapters of this book.