  We use analytic bootstrap techniques for a CFT with an interface or a
boundary. Exploiting the analytic structure of the bulk and boundary conformal
blocks we extract the CFT data. We further constrain the CFT data by applying
the equation of motion to the boundary operator expansion. The method presented
in this paper is general, and it is illustrated in the context of perturbative
Wilson-Fisher theories. In particular, we find constraints on the OPE
coefficients for the conformal interface CFT in $4 - \epsilon$ dimensions (upto
order $\mathcal{O}(\epsilon^2)$) with $\phi^4$-interactions in the bulk. We
also compute the corresponding coefficients for the non-unitary $\phi^3$-theory
in $6 - \epsilon$ dimensions in the presence of a conformal boundary equipped
with either Dirichlet or Neumann boundary conditions upto order
$\mathcal{O}(\epsilon)$, or an interface upto order
$\mathcal{O}(\sqrt{\epsilon})$.
