Outlier-contaminated normal errors in regression problems are modelled by exponential power distributions and the resulting maximum likelihood estimators are shown to involveLpminimisations (1 <p≤ 2). It is shown thatL1 +estimation is minimax outlier-robust and minimax covariance-robust over the neighbourhood of exponential power distributions. Efficiency loss is negligible. Recursive gradient-typeLpestimators are derived and shown to be convergent and consistent. The major limitation on outlier robustness is seen to be the requirement for convergence of the recursive minimisation. The algorithm is validated with an application in adaptive control.