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I will describe recent progress in the study of loop-level scattering amplitudes in gauge theory and gravity via ambitwistor strings and the scattering equations. Ambitwistor strings are worldsheet models of quantum field theories, inspired by string theory. They naturally lead to a representation of amplitudes based on the scattering equations. I will describe how the usual loop-level string formalism of higher-genus Riemann surfaces can be avoided, by turning a higher-genus surface into a nodal Riemann sphere, with the nodes representing the loop momenta. This leads to new formulas for amplitudes in gauge theory and gravity, expressed in terms of the loop-level scattering equations. I will also discuss more recent results showing how to express the one-loop integrands explicitly, without the assistance of the scattering equations, in a manner that exhibits the colour-kinematics duality and the double copy between gauge theory and gravity.