  An algebraic approach to integrable quantum field theory with a boundary (a
half line) is presented and interesting algebraic equations, Reflection
equations (RE) and Reflection Bootstrap equations (RBE) are discussed. The
Reflection equations are a consistent generalisation of Yang-Baxter equations
for factorisable scatterings on a half line (or with a reflecting boundary).
They determine the so-called reflection matrices. However, for Toda field
theory and/or other theories with diagonal S-matrices, the Reflection-Bootstrap
equations proposed by Fring and K\"oberle determine the reflection matrices,
since the reflection equations and the Yang-Baxter equations become trivial in
these cases. The explicit forms of the reflection matrices together with their
symmetry properties are given for various Toda field theories, simply laced and
non-simply laced.
