  We introduce and study higher genera generalizations of the Halphen theory of
continued fractions. The basic notion we start with is hyperelliptic Haplhen
(HH) element $$\frac{\sqrt{X_{2g+2}}-\sqrt{Y_{2g+2}}}{x-y},$$ depending on
parameter $y$, where $X_{2g+2}$ is a polynomial of degree $2g+2$ and
$Y_{2g+2}=X_{2g+2}(y)$. We study regular and irregular HH elements. their
continued fraction development and some basic properties of such development:
even and odd symmetry and periodicity.
