Recent technical advances in ultrafast laser sources enable the generation of femtosecond and attosecond soft X-ray pulses in tabletop laser setups as well as accelerator-based synchrotron and free-electron laser sources. These new light sources can be harnessed via pump-probe spectroscopy to elucidate ultrafast quantum dynamics in atoms, molecules, and condensed matter with unprecedented time resolution and chemical sensitivity. Employing such ultrashort pulses in transient X-ray absorption spectroscopy combines the unique advantages of core-level absorption probing of chemical environments and oxidation states with the ability to obtain ultimately freeze-frame snapshots of electronic and nuclear dynamics. In this Perspective, we provide an overview of the progress in applying the recently developed technique of femtosecond to attosecond time-resolved soft X-ray transient absorption spectroscopy to the study of ultrafast phenomena, including some of our own efforts to elucidate the interaction of intense laser pulses with atoms and molecules in the strong-field, nonperturbative limit. Possible avenues for future work are outlined.