  In this paper, we use the theory of genus fields to study the Euclidean
ideals of certain real biquadratic fields $K.$ Comparing with the previous
works, our methods yield a new larger family of real biquadratic fields $K$
having Euclidean ideals; and the conditions for our family seem to be more
efficient for the computations. Moreover, the previous approaches mainly focus
on the case if $h_K=2$, while the present approach can also deal with the
general case when $h_K=2^t (t\geq1)$, where $h_K$ denotes the ideal class
number of $K$. In particular, if $h_K\geq 4$, it shows that $H(K)$, the Hilbert
class field of $K$, is always non-abelian over $\mathbb{Q}$ for the family of
$K$ given in this paper having Euclidean ideals, whereas the previous
approaches always requires that $H(K)$ is abelian over $\mathbb{Q}$ explicitly
or implicitly. Finally, some open questions have also been listed for further
research.
