Computation off-loading in mobile edge computing (MEC) systems constitutes an efficient paradigm of supporting resource-intensive applications on mobile devices. However, the benefit of MEC cannot be fully exploited, when the communications link used for off-loading computational tasks is hostile. Fortunately, the propagation-induced impairments may be mitigated by intelligent reflecting surfaces (IRS), which are capable of enhancing both the spectral- and energy-efficiency. Specifically, an IRS comprises an IRS controller and a large number of passive reflecting elements, each of which may impose a phase shift on the incident signal, thus collaboratively improving the propagation environment. In this paper, the beneficial role of IRSs is investigated in MEC systems, where single-antenna devices may opt for off-loading a fraction of their computational tasks to the edge computing node via a multi-antenna access point with the aid of an IRS. Pertinent latency-minimization problems are formulated for both single-device and multi-device scenarios, subject to practical constraints imposed on both the edge computing capability and the IRS phase shift design. To solve this problem, the block coordinate descent (BCD) technique is invoked to decouple the original problem into two subproblems, and then the computing and communications settings are alternatively optimized using low-complexity iterative algorithms. It is demonstrated that our IRS-aided MEC system is capable of significantly outperforming the conventional MEC system operating without IRSs. Quantitatively, about $20~\%$ computational latency reduction is achieved over the conventional MEC system in a single cell of a $300~\rm{m}$ radius and $5$ active devices, relying on a $5$-antenna access point.