Vibrational energy relaxation and vibrational cooling of polyatomic liquids were studied with the ultrafast infrared-Raman (IR-Raman) technique. In the IR-Raman technique, a type of two-dimensional vibrational spectroscopy, a vibrational transition is pumped with a mid-infrared pulse and the instantaneous populations of all Raman-active transitions are simultaneously probed via incoherent anti-Stokes Raman scattering of a time-delayed visible pulse. The theoretical framework for these measurements, including force-force correlation function methods and perturbative techniques, is reviewed. Experimental aspects of the IR-Raman technique are discussed, including laser instrumentation, experimental set-up, the nature of the pumping and probing processes, detection sensitivity and optical background, and the interpretation of results including spectroscopic artifacts. Then examples are provided from recent research by our group, focusing on timely problems such as the pseudo-vibrational cascade, the dynamics of doorway vibrations, dynamics of overtones with Fermi resonance, multiple vibrational excitations via combination band pumping and spectral evolution in associated liquids.