Thesis advisor: Avner D. Ash / We analyze the overconvergent cohomology modules introduced by Ash and Stevens, applying them to construct eigenvarieties for fairly general reductive groups. We then establish several instances of p-adic Langlands functoriality for these eigenvarieties, and we use these functorialities to give evidence for a precise conjecture relating trianguline Galois representations to overconvergent cohomology classes. Our main technical innovations are a family of universal coefficients spectral sequences for overconvergent cohomology and a generalization of Chenevier's interpolation theorem. / Thesis (PhD) — Boston College, 2013. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.