  The positive velocity shift of absorption transitions tracing diffuse
material observed in a galaxy spectrum is an unambiguous signature of gas flow
toward the host system. Spectroscopy probing, e.g., NaI D resonance lines in
the rest-frame optical or MgII and FeII in the near-ultraviolet is in principle
sensitive to the infall of cool material at temperatures ~ 100-10,000 K
anywhere along the line of sight to a galaxy's stellar component. However,
secure detections of this redshifted absorption signature have proved
challenging to obtain due to the ubiquity of cool gas outflows giving rise to
blueshifted absorption along the same sightlines. In this chapter, we review
the bona fide detections of this phenomenon. Analysis of NaI D line profiles
has revealed numerous instances of redshifted absorption observed toward
early-type and/or AGN-host galaxies, while spectroscopy of MgII and FeII has
provided evidence for ongoing gas accretion onto >5% of luminous, star-forming
galaxies at z ~ 0.5-1. We then discuss the potentially ground-breaking benefits
of future efforts to improve the spectral resolution of such studies, and to
leverage spatially-resolved spectroscopy for new constraints on inflowing gas
morphology.
