We build up local, time translation covariant Boundary Quantum Field Theory nets of von Neumann algebras A_V on the Minkowski half-plane M_+ starting with a local conformal net A of von Neumann algebras on the real line and an element V of a unitary semigroup E(A) associated with A. The case V=1 reduces to the net A_+ considered by Rehren and one of the authors; if the vacuum character of A is summable A_V is locally isomorphic to A_+. We discuss the structure of the semigroup E(A). By using a one-particle version of Borchers theorem and standard subspace analysis, we provide an abstract analog of the Beurling-Lax theorem that allows us to describe, in particular, all unitaries on the one-particle Hilbert space whose second quantization promotion belongs to of E(A^(0)) with A^(0) the U(1)-current net. Each such unitary is attached to a scattering function or, more generally, to a symmetric inner function. We then obtain families of models via any Buchholz-Mach-Todorov extension of A^(0). A further family of models comes from the Ising model.