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Starting with Witten's twistor string, chiral string theories have emerged that describe field theory amplitudes without the towers of massive states of conventional strings. These models are known as ambitwistor strings due to their target space; the space of complexified null geodesics, also called ambitwistor space. Correlators in these string theories directly yield compact formulae for tree-level amplitudes and loop integrands, in the form of worldsheet integrals fully localized on solutions to constraints known as the scattering equations. In this chapter, we discuss two incarnations of the ambitwistor string: a 'vector representation' starting in space-time and structurally resembling the RNS superstring, and a four-dimensional twistorial version closely related to, but distinct from Witten's original model. The RNS-like models exist for several theories, with 'heterotic' and type II models describing super-Yang-Mills and 10d supergravities respectively, and they manifest the double copy relations directly at the level of the worldsheet models. In the second half of the chapter, we explain how the underlying models lead to diverse applications, ranging from extensions to new sectors of theories, loop amplitudes and to scattering on curved backgrounds. We conclude with a brief discussion of connections to conventional strings and celestial holography.