  Spatial gaps correspond to the projection in position space of the gaps of a
periodic structure whose envelope varies spatially. They can be easily
generated in cold atomic physics using finite-size optical lattice, and provide
a new kind of tunnel barriers which can be used as a versatile tool for quantum
devices. We present in detail different theoretical methods to quantitatively
describe these systems, and show how they can be used to realize in one
dimension matter wave Fabry-Perot cavities. We also provide experimental and
numerical results that demonstrate the interest of spatial gaps structures for
phase space engineering. We then generalize the concept of spatial gaps in two
dimensions and show that this enables to design multiply connected cavities
which generate a quantum dot structure for atoms or allow to construct curved
wave guides for matter waves. At last, we demonstrate that modulating in time
the amplitude of the periodic structure offers a wide variety of possible atom
manipulations including the control of the scattering of an incoming wave
packet, the loading of cavities delimited by spatial gaps, their coupling by
multiphonon processes or the realization of a tunable source of atoms. This
large range of possibilities offered by space and time engineering of optical
lattices demonstrates the flexibility of such band gap structures for matter
wave control, quantum simulators and atomtronics.
