  State-space models can be used to incorporate subject knowledge on the
underlying dynamics of a time series by the introduction of a latent Markov
state-process. A user can specify the dynamics of this process together with
how the state relates to partial and noisy observations that have been made.
Inference and prediction then involves solving a challenging inverse problem:
calculating the conditional distribution of quantities of interest given the
observations. This article reviews Monte Carlo algorithms for solving this
inverse problem, covering methods based on the particle filter and the ensemble
Kalman filter. We discuss the challenges posed by models with high-dimensional
states, joint estimation of parameters and the state, and inference for the
history of the state process. We also point out some potential new developments
which will be important for tackling cutting-edge filtering applications.
