  We determine the structure of two variations on the Temperley-Lieb algebra,
both used for dealing with special kinds of boundary conditions in statistical
mechanics models.
  The first is a new algebra, the `blob' algebra (the reason for the name will
become obvious shortly!). We determine both the generic and all the exceptional
structures for this two parameter algebra. The second is the periodic
Temperley-Lieb algebra. The generic structure and part of the exceptional
structure of this algebra have already been studied. Here we complete the
analysis, using results from the study of the blob algebra.
