  A full rational CFT, consistent on all orientable world sheets, can be
constructed from the underlying chiral CFT, i.e. a vertex algebra, its
representation category C, and the system of chiral blocks, once we select a
symmetric special Frobenius algebra A in the category C [I]. Here we show that
the construction of [I] can be extended to unoriented world sheets by
specifying one additional datum: a reversion on A - an isomorphism from the
opposed algebra of A to A that squares to the twist. A given full CFT on
oriented surfaces can admit inequivalent reversions, which give rise to
different amplitudes on unoriented surfaces, in particular to different Klein
bottle amplitudes.
  We study the classification of reversions, work out the construction of the
annulus, Moebius strip and Klein bottle partition functions, and discuss
properties of defect lines on non-orientable world sheets. As an illustration,
the Ising model is treated in detail.
