Stochastic allocation of resources in the context of wireless systems ultimately demands reactive decision making for meaningfully optimizing network-wide random utilities, while respecting certain resource constraints. Standard ergodic-optimal policies are however susceptible to the statistical variability of fading, often leading to systems which are severely unreliable and spectrally wasteful. On the flip side, minimax/outage-optimal policies are too pessimistic and often hard to determine. We propose a new risk-aware formulation of the resource allocation problem for standard multi-user point-to-point power-constrained communication with no cross-interference, by employing the Conditional Value-at-Risk (CV@R) as a measure of fading risk. A remarkable feature of this approach is that it is a convex generalization of the ergodic setting while inducing robustness and reliability in a fully tunable way, thus bridging the gap between the (naive) ergodic and (conservative) minimax approaches. We provide a closed-form expression for the CV@R-optimal policy given primal/dual variables, extending the classical stochastic waterfilling policy. We then develop a primal-dual tail-waterfilling scheme to recursively learn a globally optimal risk-aware policy. The effectiveness of the approach is verified via detailed simulations.