We will explore the possibility of getting uniform spectral gaps for some invariant differential operators on hyperbolic manifolds. We’ll begin with some basics of hyperbolic geometry, relate the spectra of hyperbolic manifolds to representation theory, and explain the term `tempered gap’. We’ll then construct a sequence of spin hyperbolic surfaces with a uniform spectral gap for the Dirac operator and a sequence of hyperbolic 3-folds with a uniform gap for the Hodge Laplacian on coclosed 1-forms. Based on joint works with A. Abdurrahman, A. Adve, B. Lowe, and J. Zung.