Accurately characterizing the spectral properties of environmental noise in open quantum systems is both a prerequisite for quantitative dynamical modeling and prediction, and a key task toward validating and optimizing control performance within fault-tolerant quantum information processing. In this talk, I will describe how open-loop quantum control may be used to devise quantum noise spectroscopy protocols for reconstructing the power spectra of the noise that qubits experience out of measurable quantities. In particular, I will focus on recent results showing how noise spectroscopy may be pushed beyond the simplest setting of single-qubit classical Gaussian dephasing, allowing for both non-Gaussian spectroscopy of classical and quantum noise, as well as complete cross-correlation spectroscopy in principle.