Since the landmark work of R. E. Kalman in the 1960s, considerable efforts have been devoted to time series state space models for a large variety of dynamic estimation problems. In particular, parametric filters that seek exact analytical estimates based on closed-form Markov-Bayes recursion, e.g., recursion from a Gaussian or gaussian mixture (GM) prior to a Gaussian/GM posterior (termed Gaussian conjugacy in this paper), form the backbone for general time series filter design. Due to challenges arising from nonlinearity, multimode (including target maneuver), intractable uncertainties (such as unknown inputs and/or non-Gaussian noises) and constraints (including circular quantities), and so on, new theories, algorithms and technologies are continuously being developed in order to maintain, or approximate to be more precise, such a conjugacy. They have in a large part contributed to the prospective developments of time series parametric filters in the last six decades. This paper reviews the stateof- the-art in distinctive categories and highlights some insights which may otherwise be overlooked. In particular, specific attention is paid to nonlinear systems with very informative observation, multimodal systems including gaussian mixture posterior and maneuvers, intractable unknown inputs and constraints, to fill the voids in existing reviews/surveys. To go beyond a pure review, we also provide some new thoughts on alternatives to the first order Markov transition model and filter evaluation with computing complexity.