Real life problems are full of uncertainty. How we handle it is important, since it affects the design and performance of algorithms. Often, the uncertainty is assumed to follow some known distribution, but in practice the estimate of the distribution may or may not be accurate. At other times, the uncertainty is assumed to be adversarial, but this can be too pessimistic for most real life instances. Advice-augmented algorithms aim to bridge the gap between these two models. In this framework, the algorithm is given some advice or prediction (e.g. from historical data, forecasts, or expert advice), whose quality is unknown. We aim to design algorithms that perform well when the quality is high (consistency), yet remain robust in their performance even when the quality is low (robustness). We consider advice-augmented algorithms for two problems. The first is two-stage matching: We design an algorithm that attains the optimal tradeoff between consistency and robustness. The second is Nash social welfare maximization in online resource allocation: We show that access to reasonable predictions gives an exponential improvement over the worst-case performance. Convex optimization plays a key role in both results.