Comment: 39 pages, 4 figures, 3 tablesThe classical tests in the instrumental variable model can behave arbitrarily if the data is contaminated. For instance, one outlying observation can be enough to change the outcome of a test. We develop a framework to construct testing procedures that are robust to weak instruments, outliers and heavy-tailed errors in the instrumental variable model. The framework is constructed upon M-estimators. By deriving the influence functions of the classical weak instrument robust tests, such as the Anderson-Rubin test, K-test and the conditional likelihood ratio (CLR) test, we prove their unbounded sensitivity to infinitesimal contamination. Therefore, we construct contamination resistant/robust alternatives. In particular, we show how to construct a robust CLR statistic based on Mallows type M-estimators and show that its asymptotic distribution is the same as that of the (classical) CLR statistic. The theoretical results are corroborated by a simulation study. Finally, we revisit three empirical studies affected by outliers and demonstrate how the new robust tests can be used in practice.