  Continuum models are particularly appealing for theoretical studies of bound
states, due to simplicity of their bulk Hamiltonians. The main challenge on
this path is a systematic description of the boundary, which comes down to
determining proper boundary conditions (BCs). BCs are a consequence of the
fundamental principle of quantum mechanics: norm conservation of the wave
function, which leads to the conservation of the probability current at the
boundary. The notion of {\em general BCs} arises, as a family of all possible
BCs that satisfy the current-conservation principle. Ahari, Ortiz, and Seradjeh
formulated a systematic derivation procedure of the general BCs from the
current-conservation principle for the 1D Hamiltonian of the most general form.
The procedure is based on the diagonalization of the current and leads to the
universal ``standardized'' form of the general BCs, parameterized in a
nonredundant one-to-one way by unitary matrices. In this work, we substantiate,
elucidate, and expand this {\em formalism of general boundary conditions for
continuum models}, addressing in detail a number of important physical and
mathematical points. We provide a detailed derivation of the general BCs from
the current-conservation principle and establish the conditions for when they
are admissible in the sense that they describe a well-defined boundary, which
is directly related to a subtle but crucial distinction between self-adjoint
(hermitian) and only symmetric operators. We provide a natural physical
interpretation of the structure of the general BCs as a scattering process and
an essential mathematical justification that the formalism is well-defined for
Hamiltonians of momentum order higher than linear. We discuss the physical
meaning of the general BCs and outline the application schemes of the
formalism, in particular, for the study of bound states in topological systems.
