How does the Kalman filter work and where is it applied?

The Kalman filter optimally estimates system states from noisy measurements using a predict-correct cycle. Prediction: project state and covariance forward using system model. Correction: update prediction using new measurement, weighted by Kalman gain K (balances model uncertainty vs measurement noise). K = P*H'*(H*P*H' + R)^(-1). The filter minimizes mean-squared estimation error. Requires known process (Q) and measurement (R) noise covariances. Extensions include Extended KF (linearizes nonlinear systems) and Unscented KF (better nonlinear approximation). Ubiquitous in navigation, tracking, and sensor fusion.