  We use Riordan array theory to give characterizations of the Borel triangle
and its associated polynomial sequence. We show that the Borel polynomials are
the moment sequence for a family of orthogonal polynomials whose coefficient
array is a Riordan array. The role of the Catalan matrix in defining the Borel
triangle is examined. We generalize the Borel triangle to a family of two
parameter triangles. Generating functions are expressed as Jacobi continued
fractions, as well as the zeros of appropriate quadratic expressions. The Borel
triangle is exhibited as a Hadamard product of matrices. We investigate the
reversions of the triangles studied. We introduce the notion of Fuss-Borel
triangles and Fuss-Catalan triangles. We end with some remarks on the Catalan
triangle.
