Measurement-based quantum computation is unusual among quantum computational models in that it does not have an obvious classical analogue. In this talk, I shall describe some new results which shed some new light on this. In the one-way model [1], computation proceeds by adaptive single-qubit measurements on a multi-qubit entangled \'cluster state\'. The adaptive measurements require a classical computer, which processes the previous measurement outcomes to determine the correct bases for the following measurement. We shall describe a generalisation of the model where this classical \'side-computation\' plays a more central role. We shall show that this classical computer need not be classically universal, and can instead by performed by a limited power \'CNOT computer\' - a reversible classical computer whose generating gate set consists of CNOT and NOT. The CNOT computer is not universal for classical computation and is believed to be less powerful. Most notably in the context of quantum computation, it is the class of computer sufficient for the efficient simulation of Clifford group circuits [2] - a closely related result. This motivates the question of what resource states would be universal for classical computation, if the control computer is in the CNOT class. We shall answer this question with several examples. Leading from these examples, a natural question is thus, is \'classically universal measurement based computation\' possible with solely classical physics? By considering different settings, we shall answer this question both in the negative and positive, and draw some striking connections with some well-known techniques from models of generalised no-signalling theories. [1] R. Raussendorf and H.J. Briegel, Phys Rev Lett (2001) 86 5188 [2] S. Aaronson and D. Gottesman, Phys Rev A (2004) 70 052328 This is joint work with Janet Anders. We would like to acknowledge inspiring and fruitful discussions with Hans Briegel, Akimasa Miyake, Robin Blume Kohout and Debbie Leung.