The function and performance of networks rely on their robustness, defined as their ability to continue functioning in the face of damage (targeted attacks or random failures) to parts of the network. Prior research has proposed a variety of measures to quantify robustness and various manipulation strategies to alter it. In this paper, our contributions are twofold. First, we critically analyze various robustness measures and identify their strengths and weaknesses. Our analysis suggests natural connectivity, based on the weighted count of loops in a network, to be a reliable measure. Second, we propose the first principled manipulation algorithms that directly optimize this robustness measure, which lead to significant performance improvement over existing, ad-hoc heuristic solutions. Extensive experiments on real-world datasets demonstrate the effectiveness and scalability of our methods against a long list of competitor strategies.