  Given an element $f$ in a regular local ring, we study matrix factorizations
of $f$ with $d \ge 2$ factors, that is, we study tuples of square matrices
$(\varphi_1,\varphi_2,\dots,\varphi_d)$ such that their product is $f$ times an
identity matrix of the appropriate size. Several well known properties of
matrix factorizations with $2$ factors extend to the case of arbitrarily many
factors. For instance, we show that the stable category of matrix
factorizations with $d\ge 2$ factors is naturally triangulated and we give
explicit formula for the relevant suspension functor. We also extend results of
Kn\"orrer and Solberg which identify the category of matrix factorizations with
the full subcategory of maximal Cohen-Macaulay modules over a certain
non-commutative algebra $\Gamma$. As a consequence of our findings, we observe
that the ring $\Gamma$ behaves, homologically, like a "non-commutative
hypersurface ring" in the sense that every finitely generated module over
$\Gamma$ has an eventually $2$-periodic projective resolution.
