The Hodrick-Prescott (HP) method was originally developed for the smoothing of time series, i.e. to get a smooth (long-term) component. We show that the HP smoother can be viewed as a Bayesian linear model with a strong prior for the smoothness component. Extending this Bayesian approach in a linear model set-up is possible by a conjugate and a non-conjugate model using MCMC. The Bayesian HP smoothing model is also extended to a spatial smoothing model. We have to define spatial neighbors for each observation and we can use in a similar way a smoothness prior as for the HP filter in time series. The new smoothing approaches are applied to the (textbook) Airline passenger data in the time series context and the problem of smoothing spatial regional data. This new approach can be used for a new class of model-based smoothers for time series and spatial models.