Underwater acoustic channels are characterized by limited bandwidth, long multipath delay spread, and severe time variation, which make reliable and high-rate communication challenging. Channel-estimate-based equalization is a key technique for compensating for distortions introduced by the channel. In this paper, low complexity algorithms for estimation and equalization of doubly spread underwater acoustic channels are presented. By exploiting the sparsity in the delay-Doppler domain, a fast projected gradient method (FPGM) is developed for estimating the delay-Doppler spread function of a time-varying channel. The FPGM formulates sparse channel estimation as a complex-valued convex optimization using an ℓ1-norm constraint. Unlike the conventional methods that split the complex variables into their real and imaginary parts, the FPGM directly handles the complex variables as a whole. A unified framework, which includes the time-reversal, linear MMSE, and decision feedback equalizer, is also proposed for fast equalization of doubly spread channels. By exploiting the special block Toeplitz-like structure of the coefficient matrix, the computational complexity of channel estimation and equalization is on the order of LNlog N, where L is the dimension of the Doppler shift and N is the signal length. [Work supported by Chinese 863 high-tech program under Grant 2009AA093601]