  We study the notion of $\Gamma$-graded commutative algebra for an arbitrary
abelian group $\Gamma$. The main examples are the Clifford algebras already
treated by Albuquerque and Majid. We prove that the Clifford algebras are the
only simple finite-dimensional associative graded commutative algebras over
$\mathbb{R}$ or $\mathbb{C}$. Our approach also leads to non-associative graded
commutative algebras extending the Clifford algebras.
