Microorganisms are able to respond effectively to diverse signals from their environment and internal metabolism because they possess a sophisticated information processing capacity. A central aim of synthetic biology is to control and reprogramme the signal processing pathways within living cells so as to realise repurposed, beneficial applications ranging from disease diagnosis and environmental sensing to chemical bioproduction. Up to now most examples of synthetic biological signal processing have been built based on digital information flow, though analogue computing is being developed to cope with more complex operations and larger sets of variables. Great progress has been made in expanding the categories of characterised biological components that can be used for cellular signal manipulation, thereby allowing synthetic biologists to more rationally programme increasingly complex behaviours into living cells. Here we provide an overview of the components and strategies that exist for designer cell signal processing and decision making, discuss how these have been implemented in prototype systems for therapeutic, environmental, and industrial biotechnological applications, and examine emerging challenges in this promising field.