We begin with an overview of the scattering equations, CHY formalism and the ambitwistor string. We discuss one of the striking simplifications that occur upon restricting to four spacetime dimensions, namely that the scattering equations with $n$ particles decompose into sectors, graded by an integer $1\le d\le n-3$. It is a non-trivial fact that the dimension agnostic CHY formulae reduce to twistor formulae once the external kinematics is restricted to four dimensions. To establish the link between them, we find and prove a formula which describes the splitting of 4-vector-valued fermion correlators on the sphere into a product of two terms, each involving left/right-handed spinors only.

We use this splitting result to derive a formula for $ \mathcal{N} = 4 $ super Einstein-Yang-Mills in twistor space based on the refined 4d scattering equations. It computes all tree level amplitudes, in all trace sectors, of minimally coupled sEYM with one gluon multiplet and two, CP conjugate, gravity multiplets. The RSV formula for \Neqfour super-Yang-Mills and a certain subsector of the CS formula for \Neqeight super-gravity are shown to be contained as special cases.

Next, we return to the dimension agnostic setting and present a collection of new ambitwistor string models, which compute the CHY formulae for DBI, Galileons, and several other low energy effective field theories. We describe two attempts at constructing an ambitwistor string for Einstein-Yang-Mills, and why they fail.

In chapter II we initiate a study of the ambitwistor string on a group manifold. After studying the classical theory and quantization on a generic group manifold, we specialize to an $AdS_3 \times S^3 $ background with pure NS-NS flux. We describe how the quantum consistency of the model requires the background and fluctuations to satisfy the supergravity equations of motion, construct explicit vertex operators and discuss correlators. We explore the prospect of a localisation on generalised scattering equations.

In chapter III we present a new operator in the ambitwistor string CFT which allows the computation of amplitudes by gluing together correlators with fewer points or of lower genus. We conjecture it to be the infinite tension limit of the standard string propagator. Due to the finiteness of the ambitwistor string spectrum, the gluing operator turns out to be a tractable object.

We demonstrate by explicit calculations how our operator underpins the recursive construction of tree-level CHY scattering amplitudes by Dolan & Goddard, as well as the computation of loop integrands on a Riemann sphere by Geyer et al. The gluing operator is schematically a product of two standard ambitwistor vertex operators. It is suitably continued to off-shell momentum while retaining BRST invariance, and intuitively represents the target space Feynman propagator.

We conclude with an exposition of some unsolved problems, open questions, new ideas and aspirations.