The Tsuji–Trost reaction (also called the Trost allylic alkylation or allylic alkylation) is a palladium-catalysed substitution reaction involving a substrate that contains a leaving group in an allylic position. The palladium catalyst first coordinates with the allyl group and then undergoes oxidative addition, forming the -allyl complex. This allyl complex can then be attacked by a nucleophile, resulting in the substituted product.
This work was first pioneered by Jirō Tsuji in 1965 and, later, adapted by Barry Trost in 1973 with the introduction of phosphine ligands. 
The scope of this reaction has been expanded to many different carbon, nitrogen, and oxygen-based nucleophiles, many different leaving groups, many different phosphorus, nitrogen, and sulfur-based ligands, and many different metals (although palladium is still preferred). 
The introduction of phosphine ligands led to improved reactivity and numerous asymmetric allylic alkylation strategies. Many of these strategies are driven by the advent of chiral ligands, which are often able to provide high enantioselectivity and high diastereoselectivity under mild conditions. This modification greatly expands the utility of this reaction for many different synthetic applications. The ability to form carbon-carbon, carbon-nitrogen, and carbon-oxygen bonds under these conditions, makes this reaction very appealing to the fields of both medicinal chemistry and natural product synthesis.