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We study the channel equalizer (CE) based on independent component analysis (ICA) for optical orthogonal-frequency-division-multiplexing (OFDM) systems without using training symbols (TSs). We begin by studying mathematical ICA models of optical OFDM systems and considering both the direct-detection optical OFDM (DDO-OFDM) and polarization division multiplexing coherent-detection optical OFDM (PDM-CO-OFDM) systems. One purpose of this paper is to provide a comprehensive study of ICA-based CE in both DDO-OFDM and CO-OFDM systems. Next, we propose a channel matrix initialization method to solve the permutation indeterminacy and complex uncertain scaling problems in ICA for both DDO-OFDM and PDM-CO-OFDM systems. Several algorithms are then investigated for ICA-based CEs including maximization of negentropy (MN), maximum likelihood (ML), minimization of mutual information (MMI), and fast ICA. It is found that the ICA-based CEs using MN, ML, and MMI can successfully recover the OFDM signal. Through both simulation and experiment, we show that in comparison to conventional TSs-based CEs with and without inter-symbol frequency-domain averaging (ISFA) and adaptive decision-directed CE, ICA-based CEs can provide slightly better or similar performances for both SP and DP optical OFDM systems with QPSK and 16-QAM modulations. Specifically, apart from higher spectral efficiency, the ICA-based CEs show significant better chromatic dispersion and polarization mode dispersion (PMD) tolerances than TSs-based CEs with ISFA in PDM-CO-OFDM systems over long-haul transmission.