For a natural number k, Cooke introduced k-stage euclidean rings as a generalization of classical Euclidean rings.  His setting was entirely commutative. Later Leutbecher introduced them in a noncommutative settings but his aim was also studying commutative rings.   We will introduce our quasi-Euclidean rings (QE rings, for short) via matrices and continuant polynomials.  After giving some applications to decomposition of certain  2x2 singular matrices into products of idempotents, we will give different characterizations of the QE rings.  In particular we will show that Unit regular rings are QE.   Questions of left and right symmetry of the notion will be also analyzed.  If time permits we will introduce QE modules and analyze some of their particularities.  Using this setting we will give sufficient conditions for a Morita context to be QE.