  We investigate consequences of adding irrelevant (or less relevant) boundary
operators to a (1+1)-dimensional field theory, using the Ising and the boundary
sine-Gordon model as examples. In the integrable case, irrelevant perturbations
are shown to multiply reflection matrices by CDD factors: the low-energy
behavior is not changed, while various high-energy behaviors are possible,
including ``roaming'' RG trajectories. In the non-integrable case, a Monte
Carlo study shows that the IR behavior is again generically unchanged, provided
scaling variables are appropriately renormalized.
