In this talk I will discuss universal statistical properties of spectra in 4d conformal supersymmetric Yang-Mills (SYM) theories and show how they give insight into the nature of the underlying model. We will see integrability manifest itself in the planar spectra of certain SYM theories, while spectra in more generic SYM theories can be described by random matrix theory, indicating the quantum-chaotic nature of the corresponding model. In the case of the imaginary-beta deformation of N=4 SYM theory, this provides a weak-coupling analogue of the chaotic dynamics seen for classical strings in the dual background.