In this survey talk, we will review aspects of analytic torsion in real and complex geometry. Analytic torsion was introduced by Ray and Singer as a spectral invariant (a determinant) of the Hodge Laplacian in real and complex geometry. It has been used to define metrics on certain line bundles, the determinants of the cohomology. In real geometry, Ray and Singer conjectured that analytic torsion coincides with Reidemeister torsion, a combinatorial invariant. In 1978, the conjecture was established by Cheeger and M¨uller. In complex geometry, Ray and Singer computed explicitly the analytic torsion of elliptic curves. Quillen used analytic torsion to define a metric on the determinant of the cohomology of a holomorphic line bundle on a Riemann surface, and proved the relevant curvature theorem. These were the starting points of a considerable body of work, whose scope is to make analytic torsion a natural object in suitable theories of secondary invariants, and also to find applications of analytic torsion in topology, number theory and dynamical systems. The purpose of our talk will be to address some of the above questions.