  The purpose of this paper is to establish injectivity theorems for higher
direct image sheaves of canonical bundles twisted by pseudo-effective line
bundles and multiplier ideal sheaves. As applications, we generalize Koll'ar's
torsion freeness and Grauert-Riemenschneider's vanishing theorem. Moreover, we
obtain a relative vanishing theorem of Kawamata-Viehweg-Nadel type and an
extension theorem for holomorphic sections from fibers of morphisms to the
ambient space. Our approach is based on transcendental methods and works for
K"ahler morphisms and singular hermitian metrics with non-algebraic
singularities.
