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This paper addresses the stability issues of the least mean absolute third (LMAT) algorithm using the normalization based on the third order in the estimation error. A novel robust normalized least mean absolute third (RNLMAT) algorithm is therefore proposed to be stable for all statistics of the input, noise, and initial weights. For further improving the filtering performance of RNLMAT in different noises and initial conditions, the variable step-size RNLMAT (VSSRNLMAT) and the switching RNLMAT (SWRNLMAT) algorithms are proposed using the statistics of the estimation error and a switching method, respectively. The filtering performance of RNLMAT is improved by VSSRNLMAT and SWRNLMAT at the expense of affordable computational cost. RNLMAT with less computational complexity than other normalized adaptive filtering algorithms, can provide better filtering accuracy and robustness against impulsive noises. The steady-state performance of RNLMAT and SWRNLMAT in terms of the excess mean-square error is performed for theoretical analysis. Simulations conducted in system identification under different noise environments confirm the theoretical results and the superiorities of the proposed algorithms from the aspects of filtering accuracy and robustness against large outliers.