  We define a relative Yamabe invariant of a smooth manifold with given
conformal class on its boundary. In the case of empty boundary the invariant
coincides with the classic Yamabe invariant. We develop approximation technique
which leads to gluing theorems of two manifolds along their boundaries for the
relative Yamabe invariant. We show that there are many examples of manifolds
with both positive and non-positive relative Yamabe invariants.
