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Network slicing of multi-access edge computing (MEC) resources is expected to be a pivotal technology to the success of 5G networks and beyond. The key challenge that sets MEC slicing apart from traditional resource allocation problems is that edge nodes depend on tightly-intertwined and strictly-constrained networking, computation and storage resources. Therefore, instantiating MEC slices without incurring in resource over-provisioning is hardly addressable with existing slicing algorithms. The main innovation of this paper is Sl-EDGE, a unified MEC slicing framework that allows network operators to instantiate heterogeneous slice services (e.g., video streaming, caching, 5G network access) on edge devices. We first describe the architecture and operations of Sl-EDGE, and then show that the problem of optimally instantiating joint network-MEC slices is NP-hard. Thus, we propose near-optimal algorithms that leverage key similarities among edge nodes and resource virtualization to instantiate heterogeneous slices 7.5x faster and within 0.25 of the optimum. We first assess the performance of our algorithms through extensive numerical analysis, and show that Sl-EDGE instantiates slices 6x more efficiently then state-of-the-art MEC slicing algorithms. Furthermore, experimental results on a 24-radio testbed with 9 smartphones demonstrate that Sl-EDGE provides at once highly-efficient slicing of joint LTE connectivity, video streaming over WiFi, and ffmpeg video transcoding.