Coarse-grained models are a necessary evil in controlling, predicting, and understanding complex physical phenomena. Since kinetic descriptions of matter are rarely available experimentally, much work has been done to derive coarsened models to evolve observables that are available, yet rarely do such observables admit closed-form descriptions that are straightforward to obtain using bottom-up analytical approaches. In this talk I survey methods of data-driven system identification that utilize weak formulations of dynamical laws to extract coarse-grained equations from macroscale observables, focusing in particular on the WSINDy algorithm (Weak-form Sparse Identification of Nonlinear Dynamics). WSINDy and other variants have been useful in revealing sparse effective equations for quantities that may be available experimentally, despite noise and other corruptions from microscale effects persisting in such macroscale observations. I will demonstrate this utility by applying WSINDy to mean-field equation discovery from particle systems, partially-observed Hamiltonian systems, equilibrating cold gases, and kinetic closures for thermal radiation transport, where in many cases prior knowledge (symmetries, constraints, etc) can easily be enforced in the learned model. Lastly, I will touch on some open problems related to data-driven weak-form system identification.