In machine learning, the loss functions optimized during training often differ from the target loss that defines task performance due to computational intractability or lack of differentiability. We present an in-depth study of the target loss estimation error relative to the surrogate loss estimation error. Our analysis leads to $H$-consistency bounds, which are guarantees accounting for the hypothesis set $H$. These bounds offer stronger guarantees than Bayes-consistency or $H$-calibration and are more informative than excess error bounds. We begin with binary classification, establishing tight distribution-dependent and -independent bounds. We provide explicit bounds for convex surrogates (including linear models and neural networks) and analyze the adversarial setting for surrogates like $ρ$-margin and sigmoid loss. Extending to multi-class classification, we present the first $H$-consistency bounds for max, sum, and constrained losses, covering both non-adversarial and adversarial scenarios. We demonstrate that in some cases, non-trivial $H$-consistency bounds are unattainable. We also investigate comp-sum losses (e.g., cross-entropy, MAE), deriving their first $H$-consistency bounds and introducing smooth adversarial variants that yield robust learning algorithms. We develop a comprehensive framework for deriving these bounds across various surrogates, introducing new characterizations for constrained and comp-sum losses. Finally, we examine the growth rates of $H$-consistency bounds, establishing a universal square-root growth rate for smooth surrogates in binary and multi-class tasks, and analyze minimizability gaps to guide surrogate selection.