© Springer Nature Switzerland AG 2019. This paper reviews the Higher Order Nonlinear Units (HONUs) and their fundamental supervised sample-by-sample and batch learning algorithms for data-driven controller learning when only measured data are known about the plant. We recall recently introduced conjugate gradient batch learning for weakly nonlinear plant identification with HONUs and we compare its performance to classical Levenberg-Marquard (LM). Further, we recall recursive least square (RLS) adaptation and compare its performance to L-M learning both for plant approximation and controller tuning. Further, a model reference adaptive control (MRAC) strategy with efficient controller learning for linear and weakly nonlinear plants is proposed with static HONUs that avoids recurrent computations, and its potentials and limitations with respect to plant nonlinearity are discussed. Recently developed stability approach for recurrent HONUs and for closed control loops with linear plant and nonlinear (HONU) controller is recalled and discussed in connotation stability of the adaptive closed control loop.