  The connection between continued fractions and orthogonality which is
familiar for $J$-fractions and $T$-fractions is extended to what we call
$R$-fractions of type I and II. These continued fractions are associated with
recurrence relations that correspond to multipoint rational interpolants. A
Favard type theorem is proved for each type. We then study explicit models
which lead to biorthogonal rational functions.
