In this paper, the maximum correntropy (MC) criterion is used as the cost function in the online sequential extreme learning machine (OS-ELM) algorithm and constraint OS-ELM (COS-ELM) algorithm, generating the proposed OS-ELM based on maximum correntropy (OS-ELM-MC) and COS-ELM based on maximum correntropy (COS-ELM-MC). In comparison with OS-ELM and COS-ELM, the proposed OS-ELM-MC and COS-ELM-MC present superior performance in non-Gaussian noise environments and almost the same performance in Gaussian noise environments. As an important parameter, the hidden node number is also discussed by simulations in this paper. Simulations on the examples of Mackey-Glass (MG) chaotic time series prediction and nonlinear regression validate the efficiency of the proposed OS-ELM-MC and COS-ELM-MC.