The chemotaxis pathway in bacteria is a model for how cells process information about their environment. Specifically, the pathway allows cells to sense and swim towards nutrients and away from toxins. Its protein constituents include ligand-specific chemoreceptors, which cluster at the cell poles and initiate signal transduction to regulate flagellated rotary motors. These motors can either rotate counter-clockwise or clockwise, leading respectively to straight swimming or selection of a random new direction of the cell. We present a model for the signalling pathway which is able to explain recent in vivo data from FRET (fluorescence resonance energy transfer) measurements. Our model highlights the remarkable signalling properties emerging from the pathway, including high sensitivity to detect small changes in ligand concentration and precise adaptation to persistent stimulation. Detailed understanding of the bacterial chemotaxis pathway further allows us explain general properties of more complicated sensory systems.