The growing popularity of virtual and augmented reality communications and mathbf 360 video streaming is moving video communication systems into much more dynamic and resource-limited operating settings. The enormous data volume of mathbf 360 videos requires an efficient use of network bandwidth to maintain the desired quality of experience for the end user. To this end, we propose a framework for viewport-driven rate-distortion optimized mathbf 360 video streaming that integrates the user view navigation pattern and the spatiotemporal rate-distortion characteristics of the mathbf 360 video content to maximize the delivered user quality of experience for the given network/system resources. The framework comprises a methodology for constructing dynamic heat maps that capture the likelihood of navigating different spatial segments of a mathbf 360 video over time by the user, an analysis and characterization of its spatiotemporal rate-distortion characteristics that leverage preprocessed spatial tilling of the mathbf 360 view sphere, and an optimization problem formulation that characterizes the delivered user quality of experience given the user navigation patterns, mathbf 360 video encoding decisions, and the available system/network resources. Our experimental results demonstrate the advantages of our framework over the conventional approach of streaming a monolithic uniformly-encoded mathbf 360 video and a state-of-the-art reference method. Considerable video quality gains of 4 - 5 dB are demonstrated in the case of two popular 4K mathbf 360 videos.