real quaternions were introduced by hamilton in 1843 and for a while were the only advanced mathematics taught in some american universities. after briefly discussing some of their classical and modern applications outside number theory, we shall focus on rational quaternion algebras and their many applications to arithmetic. as sample results we have chosen to discuss the problem of representation of numbers by quadratic forms, a theorem of gross-zagier, and ramanujan graphs. our intention is to point to the scope and diversity of applications, ranging from classical to current, of quaternion algebras.