Living cells are able to produce a wide variety of biological responses when subjected to biochemical stimuli. It has become apparent that these biological responses are regulated by complex chemical reaction networks (CRNs). Unravelling the function of these circuits is a key topic of both systems biology and synthetic biology. Recent progress at the interface of chemistry and biology together with the realisation that current experimental tools are insufficient to quantitatively understand the molecular logic of pathways inside living cells has triggered renewed interest in the bottom-up development of CRNs. This builds upon earlier work of physical chemists who extensively studied inorganic CRNs and showed how a system of chemical reactions can give rise to complex spatiotemporal responses such as oscillations and pattern formation. Using purified biochemical components, in vitro synthetic biologists have started to engineer simplified model systems with the goal of mimicking biological responses of intracellular circuits. Emulation and reconstruction of system-level properties of intracellular networks using simplified circuits are able to reveal key design principles and molecular programs that underlie the biological function of interest. In this Tutorial Review, we present an accessible overview of this emerging field starting with key studies on inorganic CRNs followed by a discussion of recent work involving purified biochemical components. Finally, we review recent work showing the versatility of programmable biochemical reaction networks (BRNs) in analytical and diagnostic applications.