Attosecond transient absorption spectroscopy is a recent addition to the experimental tool set of attosecond science. This all-optical methodmeasures different observables than the previously existing techniques based on electron and ion detection and overcomes several of their limitations. We review the present state-of-the-art of attosecond transient absorption experiments and theory. Applications cover the exploration of ultrafast electron dynamics in atoms as well as in solid-state systems. As an example we discuss the observation of transiently bound electron wavepacket dynamics in helium in more detail. This example illustrates how transient absorption spectroscopy can provide information that is fundamentally inaccessible to the tech...