Numerical approximations can be regarded as statistical inference, if one interprets the solution of the numerical problem as a parameter in a statistical model whose likelihood links it to the information (‘data’) available from evaluating functions. This view is advocated by the field of Probabilistic Numerics and has already yielded two successes, Bayesian Optimization and Bayesian Quadrature. In an analogous manner, we construct a Bayesian probabilistic-numerical method for ODEs. To this end, we construct a probabilistic state space model for ODEs which enables us to borrow the machinery of Bayesian filtering. This unlocks the application of all Bayesian filters from signal processing to ODEs, which we name ODE filters. We theoretically analyse the convergence rates of the most elementary one, the Kalman ODE filter and discuss its uncertainty quantification. Lastly, we demonstrate how employing these ODE filters as forward simulators engenders new ODE inverse problem solvers that outperform its classical ’likelihood-free’ counterparts.