We investigate categorifications of linear algebra, and their applications to the construction of 4-manifold invariants, to the construction of a variety of linear algebraic structures in quantum information theory, and to the classification of certain instances of `quantum pseudo-telepathy', a phenomenon in quantum physics where two non-communicating parties can use pre-shared entanglement to perform tasks classically impossible without communicating. This thesis is divided into four chapters, closely following arXiv:1812.11933, arXiv:1609.07775, and arXiv:1801.09705. In the first chapter, we introduce semisimple 2-categories, fusion 2-categories and spherical fusion 2-categories. We prove that every finite semisimple 2-category is the 2-category of finite semisimple module categories of a multifusion category, and give examples of fusion 2-categories. In the second chapter, we construct, for each spherical fusion 2-category, a state-sum invariant of oriented singular piecewise-linear 4-manifolds, and show that these invariants generalize various previous 4-manifold invariants, including the Crane-Yetter invariant and a recent invariant of Cui. In the third chapter, we use biunitary connections in the 2-category of 2-Hilbert spaces to generate many new construction schemes for linear algebraic quantities of relevance to quantum information, including complex Hadamard matrices and unitary error bases, and we use these techniques to construct a unitary error basis which cannot be built using any previously known method. In the fourth chapter, we classify quantum isomorphic graphs in terms of Morita equivalence classes of algebras in certain monoidal categories, give examples of such algebras arising from groups of central type, and discuss various applications to the study of quantum pseudo-telepathy in the graph isomorphism game.