A growing number of microfluidic systems operate not through networks of microchannels but instead through using 2D flow fields. While the design rules for channel networks are already well-known and exposed in microfluidics textbooks, the knowledge underlying transport in 2D microfluidics remains scattered piecemeal and is not easily accessible to experimentalists and engineers. In this tutorial review, we formulate a unified framework for understanding, analyzing and designing 2D microfluidic technologies. We first show how a large number of seemingly different devices can all be modelled using the same concepts, namely flow and diffusion in a Hele-Shaw cell. We then expose a handful of mathematical tools, accessible to any engineer with undergraduate level mathematics knowledge, namely potential flow, superposition of charges, conformal transforms and basic convection–diffusion. We show how these tools can be combined to obtain a simple “recipe” that models almost any imaginable 2D microfluidic system. We end by pointing to more advanced topics beyond 2D microfluidics, namely interface problems and flow and diffusion in the third dimension. This forms the basis of a complete theory allowing for the design and operation of new microfluidic systems.