  We address the question of existence of absolutely simple abelian varieties
of dimension 2 with everywhere good reduction over quadratic fields. The
emphasis will be given to the construction of pairs $(K,C)$, where $K$ is a
quadratic number field and $C$ is a genus $2$ curve with everywhere good
reduction over $K$. We provide the first infinite sequence of pairs $(K,C)$
where $K$ is a real (complex) quadratic field and $C$ has everywhere good
reduction over $K$. Moreover, we show that the Jacobian of $C$ is an absolutely
simple abelian variety.
