The need to measure high repetition rate ultrafast processes cuts across multiple areas of science. The last decade has seen tremendous advances in the development and application of new techniques in this field, as well as many breakthrough achievements analyzing non-repetitive optical phenomena. Several approaches now provide convenient access to single-shot optical waveform characterization, including the dispersive Fourier transform (DFT) and time-lens techniques, which yield real-time ultrafast characterization in the spectral and temporal domains, respectively. These complementary approaches have already proven to be highly successful to gain insight into numerous optical phenomena including the emergence of extreme events and characterizing the complexity of laser evolution dynamics. However, beyond the study of these fundamental processes, real-time measurements have also been driven by particular applications ranging from spectroscopy to velocimetry, while shedding new light in areas spanning ultrafast imaging, metrology or even quantum science. Here, we review a number of landmark results obtained using DFT-based technologies, including several recent advances and key selected applications.