In recent years, machine learning has become an indispensable tool across various industry domains, revolutionizing the way businesses leverage data to make decisions. One of the key factors driving the success of machine learning is the explosion of big data which has enabled the development of increasingly sophisticated models. However, this era of big data has brought with it a unique set of challenges: prominently, (a) issues regarding data quality, and (b) scalability of learning algorithms. Addressing these challenges is essential in order to more effectively harness the synergy between big data and machine learning. This dissertation systematically addresses these challenges for certain foundational learning tasks: ranking from pairwise comparisons, hierarchical clustering, and online learning with bandit feedback. For each of these problems, we present an essentially-complete picture through algorithmic upper-bounds, and complementary lower-bounds for achieving adversarial robustness, high parallelizability and memory efficiency. More specifically, 1. We study the problem of robust estimation for the popular BTL model for ranking from offline data in a very general adversarial contamination setting. In this setting, we establish exact necessary and sufficient conditions for identifiability, showing that robustness is a structural property of the underlying topology itself. We further show that a popular class of comparison graphs — Erdős-Rényi graphs — is highly robust to contamination. For these graphs, we also design an estimation algorithm that can provably recover from a non-trivial corruption rate. 2. We then consider robust ranking in the online setting as a robust regret minimization problem for dueling bandits in a very general adversarial contamination setting. We design an algorithm that, assuming only the existence of a Condorcet winner, achieves low regret without any foreknowledge of the extent of corruption in the feedback. Moreover, we show that this regret is asymptotically optimal via a lower bound. 2. On the scalability front, we consider the top-k identification problem for ranking in a limited-adaptivity setting. When the preferences satisfy strong stochastic transitivity, we establish a lower bound on the sample-complexity with unrestricted adaptivity, and show that just 3 adaptive rounds are sufficient to achieve it (up to log factors). We also design more general multi-round algorithms with even lower sample complexities, establishing a non-trivial upper-bound on the round vs sample complexity tradeoff. 3. We then consider the problem of scalable minimum cost hierarchical clustering from paired similarity data, where we design highly parallel and memory-efficient algorithms in the massively parallel computing, and streaming model, respectively. Both algorithms are provably optimal, and follow as a consequence of novel structural properties that we prove about the clustering cost function itself. These properties and consequently, our algorithmic results, also extend to other maximization based objectives for this problem. 4. Lastly, we consider scalability for classical stochastic bandits in a space-bounded multi-pass streaming setting where we uncover a surprising phenomenon: increasing memory beyond a constant to any quantity that is sublinear in the number of arms has almost no effect on reducing regret in the worst case. We design a constant-memory algorithm that achieves low regret — both worst-case and instance-dependent — in a given fixed number of passes, and prove a highly technical lower bound showing it is not possible to do much better in those many passes, even when allowed additional superconstant memory.