The ensemble Kalman filter (EnKF) is a computational technique for approximate inference in spatio-temporal state-space models. It has been successfully used in many real-world nonlinear data-assimilation problems with very high dimensions, such as weather forecasting. However, the EnKF is most appropriate for linear Gaussian state-space models with known parameters. Here, we consider a broader class of hierarchical state-space models, which includes two additional layers: The parameter layer handles unknown variables that cannot be easily included in the state vector, while the transformation layer allows for non-Gaussian observations. For Bayesian inference in these models, we propose a general class of extended EnKFs, which approximate inference on the state existing Bayesian methods (e.g., Gibbs sampler or particle filter) using the EnKF or the ensemble Kalman smoother. Extended EnKFs enable approximate, computationally feasible filtering and smoothing in many high-dimensional, nonlinear, and non-Gaussian space-time models with unknown parameters. We highlight several examples including assimilation of heavy-tailed and discrete data and online or offline parameter estimation.