Infinitary Combinatorics Near Singular Cardinals

Large cardinals exhibit many combinatorial properties, some of which can consistently hold at small cardinals. This thesis examines several of these properties, focusing on their behavior at or near singular cardinals. In particular it discusses the strong and super tree properties, generalizations of the tree property that are closely linked with strongly compact and supercompact cardinals, and mutual stationarity, a property that provides an analogue of stationarity for singular cardinals.