  In gravitational theories with extra dimensions, it is argued that the
existence of a positive vacuum energy generically implies catastrophic
instability of our four-dimensional world. The most generic instability is a
decompactification transition to growth of the extra dimensions, although other
equally bad transitions may take place. This follows from simple considerations
based on the form of the potential for the size modulus of the extra
dimensions, and apparently offers a resolution of the conundrums presented by
eternal de Sitter space. This argument is illustrated in the context of string
theory with a general discussion of potentials generated by fluxes, wrapped
branes, and stringy corrections. Moreover, it is unlikely that the present
acceleration of the Universe represents an ongoing transition in a quintessence
scenario rolling towards decompactification, unless the higher dimensional
theory has a cosmological constant.
