Propagating optical fields can encode broadband signals, for which open quantum systems can be used to implement linear or nonlinear dynamic signal transformations. In this talk I will review our group's theoretical and experimental research on photonic signal processing and photonic circuits in the quantum-optical regime, which aims at diverse applications ranging from attojoule classical optical switching to continuous quantum error correction. The main technical themes of our recent work include nonlinear stochastic model reduction, coherent feedback quantum control, and the interplay of quantum fluctuations with nonlinear mean-field dynamics in driven dissipative systems.