1. Introduction and overview: mathematical strategies for filtering turbulent systems -- 2. Filtering a stochastic complex scalar: the prototype test problem -- 3. The Kalman filter for vector systems: reduced filters and a three-dimensional toy model -- 4. Continuous and discrete Fourier series and numerical discretization -- 5. Stochastic models for turbulence -- 6. Filtering turbulent signals: plentiful observations -- 7. Filtering turbulent signals: regularly spaced sparse observations -- 8. Filtering linear stochastic PDE models with instability and model error -- 9. Strategies for filtering nonlinear systems -- 10. Filtering prototype nonlinear slow-fast systems -- 11. Filtering turbulent nonlinear dynamical systems by finite ensemble methods -- 12. Filtering turbulent nonlinear dynamical systems by linear stochastic models -- 13. Stochastic parametrized extended Kalman filter for filtering turbulent signals with model error -- 14. Filtering turbulent tracers from partial observations: an exactly solvable test model -- 15. The search for efficient skillful particle filters for high-dimensional turbulent dynamical systems.