  We discuss the relationship between quaternion algebras and quadratic forms
with a focus on computational aspects. Our basic motivating problem is to
determine if a given algebra of rank 4 over a commutative ring R embeds in the
2x2-matrix ring M_2(R) and, if so, to compute such an embedding. We discuss
many variants of this problem, including algorithmic recognition of quaternion
algebras among algebras of rank 4, computation of the Hilbert symbol, and
computation of maximal orders.
