  We introduce and study in detail a special class of backward continued
fractions that represents a generalization of R\'enyi continued fractions. We
investigate the main metrical properties of the digits occurring in these
expansions and we construct the natural extension for the transformation that
generates the R\'enyi-type expansion. Also we define the associated random
system with complete connections whose ergodic behavior allows us to prove a
variant of Gauss-Kuzmin-type theorem.
