Phonological processes can be characterized as functions from input strings to output strings, and treating them as mathematical objects like this reveals properties that hold regardless of how we implement them (i.e., with rules, constraints, or other tools). For example, Chandlee (2014) found that a vast majority of phonological processes can be modelled as Strictly Local (SL) functions, which are sensitive to a window of finite size. Long-distance processes like vowel and consonant harmony are exceptions to this generalization, although a key observation is that they look local once irrelevant information is ignored. This thesis shows how such selective attention can be modelled by augmenting SL functions with autosegmental tiers (e.g., Goldsmith, 1976). A single tier is sufficient to capture individual long-distance processes, and having multiple tiers available allows us to model multiple long-distance processes simultaneously as well as interactions between local and non-local patterns. Furthermore, probabilistic variants of these tier-based functions allow for a cognitively plausible model of what Zymet (2015) calls distance-based decay. Unrestricted use of multiple tiers is, however, quite powerful and so I additionally argue that tiersets should be defined from the perspective of individual input elements (i.e., potential process targets). Each input element designates a superset-subset hierarchy of tiers and pays attention to them alone; the tiers specified by another input element are either redundant or irrelevant. Restricting tiersets in this manner has beneficial consequences for learnability as it imparts a structure onto the learner's hypothesis space that can be exploited to great effect. Furthermore, tier-based functions meeting this restriction fail to generate a number of pathological behaviours that can be characterized as subsequential functions, a type of function that has previously been offered as a model of non-local phonological processes (Heinz and Lai, 2013; Luo, 2017; Payne, 2017). In light of their empirical coverage, their comparative lack of pathological predictions, and their efficient learnability, I conclude that tier-based functions act as a more accurate characterization of long-distance phonology.