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Threshold autoregressive moving-average (TARMA) models are popular in time series analysis due to their ability to parsimoniously describe several complex dynamical features. However, neither theory nor estimation methods are currently available when the data present heavy tails or anomalous observations, which is often the case in applications. In this paper, we provide the first theoretical framework for robust M-estimation for TARMA models and also study its practical relevance. Under mild conditions, we show that the robust estimator for the threshold parameter is super-consistent, while the estimators for autoregressive and moving-average parameters are strongly consistent and asymptotically normal. The Monte Carlo study shows that the M-estimator is superior, in terms of both bias and variance, to the least squares estimator, which can be heavily affected by outliers. The findings suggest that robust M-estimation should be generally preferred to the least squares method. Finally, we apply our methodology to a set of commodity price time series; the robust TARMA fit presents smaller standard errors and leads to superior forecasting accuracy compared to the least squares fit. The results support the hypothesis of a two-regime, asymmetric nonlinearity around zero, characterised by slow expansions and fast contractions.