Spectral efficient transmission techniques are necessary and promising for future broadband wireless communications, where the quality of service (QoS) and/or max-min fair (MMF) of intended users are often considered simultaneously. In this paper, both the QoS problem and the MMF problem are investigated together for transmit beamforming in broadband multigroup multicast channels with frequency-selective fading characters. We first present a basic algorithm by directly using the results in frequency-flat multigroup multicast systems (Karipidis et al., 2008), namely, the approximation algorithms in this paper, for both problems, respectively. Due to high computational consumption nature of the approximation algorithms, two reduced-complexity algorithms for each of the two problems are proposed separately by introducing the time-frequency correlations. In addition, parameters in the new time-frequency formulations, such as the number of optimization matrix variables and the taps of the beamformer with finite impulse response (FIR) structure, can be used to make a reasonable tradeoff between computational burden and system performance. Insights into the relationship between the two problems and some analytical results of the computational complexity of the proposed algorithms are also studied carefully. Numerical simulations indicate the efficiency of the proposed algorithms.