  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.
