In this paper, three Lorentzian based robust adaptive algorithms are proposed for identifying systems in presence of impulsive noise. The first algorithm called Lorentzian adaptive filtering (LAF) is derived from a sliding window type cost function with Lorentzian norm of past errors to combat adverse effect of impulsive noise on systems. The first and second order convergence analyses of the LAF algorithm are carried out in this paper. Then, to identify sparse systems in impulsive noise environment,l0 norm penalty is introduced to the cost function of the LAF algorithm leading to a new algorithm called Lorentzian hard thresholding adaptive filtering (LHTAF) which employs hard thresholding operator with a fixed hard thresholding parameter to obtain sparse solutions. The effect of the hard thresholding operator is further analyzed, and the analysis shows that a variable hard thresholding parameter offers significant improvement in the performance of the algorithm, and this result in the final algorithm called Lorentzian variable hard thresholding adaptive filtering (LVHTAF) where the hard thresholding parameter is adjusted adaptively. Simulation results show that the LVTHAF outperforms the existing robust sparse adaptive algorithms by producing lesser steady state mean square error. © 2004-2012 IEEE.