- This work introduces a novel approximate nonlinear filtering paradigm for nonlinear non-Gaussian discrete-time systems. Hinging on the concept of conditional orthogonality of random sequences, the conditionally linear filtering approach is an extension of the classical linear filtering approach. The resulting estimator turns out to be conditionally linear with respect to the most recent measurement, given the passed observations, which is adequate for online filtering. It is shown that the classical Kalman filter and the conditionally Gaussian filter, i.e., the optimal filter for conditionally linear Gaussian systems, are special cases of the conditionally linear filter. When applied to the problem of mode estimation in jump-parameter systems with full state information, it provides an approximate nonlinear mode estimator that only requires the first two moments of the process noise, whereas the optimal filter (Wonham filter) requires the complete noise distribution. In the case of noisy measurements in jump-linear systems, the conditionally linear filtering approach lends itself to a meaningful hybrid estimator. The proposed hybrid estimator is successfully applied to the problem of gyro failure monitoring onboard spacecraft using noisy attitude quaternion measurements. Extensive Monte Carlo simulations show that the conditionally linear mode estimator outperforms the linear mode estimator. For Gaussian noises the conditionally linear filter and the optimal Wonham filter perform similarly. Moreover, for Weibull-type noises, the conditionally linear filter outperforms a Wonham filter that assumes Gaussian noises.