We introduce a novel system setup where a backscatter device operates in the presence of an intelligent reflecting surface (IRS). In particular, we study the bistatic backscatter communication (BackCom) system assisted by an IRS. The phase shifts at the IRS are optimized jointly with the transmit beamforming vector of the carrier emitter to minimize the transmit power consumption at the carrier emitter whilst guaranteeing a required BackCom performance. The unique channel characteristics arising from multiple reflections at the IRS render the optimization problem highly non-convex. Therefore, we jointly utilize the minorization-maximization algorithm and the semidefinite relaxation technique to present an approximate solution for the optimal IRS phase shift design. We also extend our analytical results to the monostatic BackCom system. Numerical results indicate that the introduction of the IRS brings about considerable reductions in transmit power, even with moderate IRS sizes, which can be translated to range increases over the non-IRS-assisted BackCom system.