This is a comprehensive review of recent work on zeta functions, periodic orbit theory and their interrelations with the Selberg trace formula. Work on the quantization of chaos is also surveyed. The emphasis is on Lie group representation theory and differential geometry. Connections to string theory are also discussed. The following sample of section headings will give an idea of the great variety of topics considered: Rank one symmetric spaces, trace formula, geodesics on symmetric spaces, representations of \(G\), the Selberg trace formula, zeta functions on symmetric spaces, functional equations, factorization, length spectra, Weyl's formula, transfer operators, billiards, quantization of chaos, Riemann zeta function, helium atom, hydrogen atom in a magnetic field, zeta functions on \(\Gamma\setminus \mathbb{H}^ 3\), Eisenstein series, bound states (Selberg conjecture and Phillips-Sarnak conjecture), ground states, Fermat groups, Hecke groups, regular octogon, noncompact quotients of symmetric spaces of rank one, the Selberg trace formula in the infinite volume case, hyperbolic lattice point problem, analytic torsion, Reidemeister torsion, eta invariants, locally symmetric spaces, three disc problem, chaotic spectroscopy, determinants and zeta functions, heat kernels on bundles. The paper contains a large set of references.