Extending Griffiths' classical theory of period mappings for compact Kähler manifolds, this book develops and applies a theory of period mappings of Hodge-de Rham type for families of open complex manifolds. The text consists of three parts. The first part develops the theory. The second part investigates the degeneration behavior of the relative Frölicher spectral sequence associated with a submersive morphism of complex manifolds. The third part applies the preceding material to the study of irreducible symplectic complex spaces. The latter notion generalizes the idea of an irreducible symplectic manifold, referred to as an irreducible hyperkähler manifold in differential geometry, to possibly singular spaces. The three parts of the work are of independent interest but intertwine nicely.