  We study the constraints of crossing symmetry and unitarity for conformal
field theories in the presence of a boundary, with a focus on the Ising model
in various dimensions. We show that an analytic approach to the bootstrap is
feasible for free-field theory and at one loop in the epsilon expansion, but
more generally one has to resort to numerical methods. Using the recently
developed linear programming techniques we find several interesting bounds for
operator dimensions and OPE coefficients and comment on their physical
relevance. We also show that the "boundary bootstrap" can be easily applied to
correlation functions of tensorial operators and study the stress tensor as an
example. In the appendices we present conformal block decompositions of a
variety of physically interesting correlation functions.
