The evolution of mobile communication networks has always been accompanied by the advancement of ISI mitigation techniques, from equalization in 2G, spread spectrum and RAKE receiver in 3G, to OFDM in 4G and 5G. Looking forward towards 6G, by exploiting the high spatial resolution brought by large antenna arrays and the multi-path sparsity of mmWave and Terahertz channels, a novel ISI mitigation technique termed delay alignment modulation (DAM) was recently proposed. However, existing works only consider the single-carrier perfect DAM, which is feasible only when the number of BS antennas is no smaller than that of channel paths, so that all multi-path signal components arrive at the receiver simultaneously and constructively. This imposes stringent requirements on the number of BS antennas and multi-path sparsity. In this paper, we propose a generic DAM technique to manipulate the channel delay spread via spatial-delay processing, thus providing a flexible framework to combat channel time dispersion for efficient single- or multi-carrier transmissions. We first show that when the number of BS antennas is much larger than that of channel paths, perfect delay alignment can be achieved to transform the time-dispersive channel to time non-dispersive channel with the simple delay pre-compensation and path-based MRT beamforming. When perfect DAM is infeasible or undesirable, the proposed generic DAM technique can be applied to significantly reduce the channel delay spread. We further propose the novel DAM-OFDM technique, which is able to save the CP overhead or mitigate the PAPR issue suffered by conventional OFDM. We show that the proposed DAM-OFDM involves joint frequency- and time-domain beamforming optimization, for which a closed-form solution is derived. Simulation results show that the proposed DAM-OFDM outperforms the conventional OFDM in terms of spectral efficiency, BER and PAPR.