We propose DANSE - a data-driven non-linear state estimation method. DANSE provides a closed-form posterior of the state of a model-free process, given linear measurements of the state in a Bayesian setup, like the celebrated Kalman filter (KF). Non-linear dynamics of the state are captured by data-driven recurrent neural networks (RNNs). The training of DANSE combines maximum-likelihood and gradient-descent in an unsupervised framework, i.e. only measurement data and no process data are required. Using simulated linear and non-linear process models, we demonstrate that DANSE - without knowledge of the process model - provides competitive performance against model-based approaches such as KF, unscented KF (UKF), extended KF (EKF), and a hybrid approach such as KalmanNet.