Nonreciprocal microwave devices are key components of communication platforms. Nonreciprocity can arise in chiral systems, where chirality refers to a fixed handedness that is preserved under time reversal. Chiral excitations (quasiparticles) provide opportunities for the realization of miniaturized microwave components with directional properties. In particular, surface acoustic waves that propagate in magnetic media are chiral and can display pronounced nonreciprocal character. Because surface acoustic waves are an established technological platform, hybrid surface acoustic wave/spin wave devices have great application potential. In this mini-review, we introduce the general concept of chiral and nonreciprocal magnetoacoustic waves. We discuss a widely employed phenomenological model based on magnetoelastic coupling and magneto-rotation that quantitatively accounts for many experimental findings and give a brief overview over selected experiments and advances in this emerging research field.