`How do our ideas about quantum mechanics affect our understanding of spacetime?' This familiar question leads to quantum gravity. The complementary question is also important: `How do our ideas about spacetime affect our understanding of quantum mechanics?' This short abstract of a talk given at the Gafka2004 conference contains a very brief summary of some of the author's papers on generalizations of quantum mechanics needed for quantum gravity. The need for generalization is motivated. The generalized quantum theory framework for such generalizations is described and illustrated for usual quantum mechanics and a number of examples to which it does not apply. These include spacetime alternatives extended over time, time-neutral quantum theory, quantum field theory in fixed background spacetime not foliable by spacelike surfaces, and systems with histories that move both forward and backward in time. A fully four-dimensional, sum-over-histories generalized quantum theory of cosmological geometries is briefly described. The usual formulation of quantum theory in terms of states evolving unitarily through spacelike surfaces is an approximation to this more general framework that is appropriate in the late universe for coarse-grained descriptions of geometry in which spacetime behaves classically. This abstract is unlikely to be clear on its own, but references are provided to the author's works where the ideas can be followed up.

8 pages, LATEX, a very brief abstract of much work