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We present a design of a complete and practical scheduler for the 3GPP Long Term Evolution (LTE) downlink by integrating recent results on resource allocation, fast computational algorithms, and scheduling. Our scheduler has low computational complexity. We define the computational architecture and describe the exact computations that need to be done at each time step (1 milliseconds). Our computational framework is very general, and can be used to implement a wide variety of scheduling rules. For LTE, we provide quantitative performance results for our scheduler for full buffer, streaming video (with loose delay constraints), and live video (with tight delay constraints). Simulations are performed by selectively abstracting the PHY layer, accurately modeling the MAC layer, and following established network evaluation methods. The numerical results demonstrate that queue- and channel-aware QoS schedulers can and should be used in an LTE downlink to offer QoS to a diverse mix of traffic, including delay-sensitive flows. Through these results and via theoretical analysis, we illustrate the various design tradeoffs that need to be made in the selection of a specific queue-and-channel-aware scheduling policy. Moreover, the numerical results show that in many scenarios strict prioritization across traffic classes is suboptimal.