We shall discuss the so-called Pell-Abel equation, i.e. X2-D(t)Y2=1. This is similar to the classical Pell equation (proposed in fact by Fermat), which appears in many issues of Number Theory. The polynomial version seems less well-known, but is old as well, studied e.g. by Abel in 1826. For instance he related the equation with a continued fraction for \sqrt{D(t)}, as in the numerical case. We shall survey about this equation, illustrating different mathematical issues involving it. We shall also mention some recent results about the structure of the continued fraction: it need not be periodic, but the sequence of degrees of the partial quotients always is.