  We explore different generalizations of the classical concept of independent
families on $\omega$ following the study initiated by Fisher and Montoya. We
show that under $\diamondsuit^*(\kappa)$ we can get strongly independent
families on $\kappa$ of size $2^\kappa$ and present an equivalence of
$\mathsf{GCH}$ in terms of strongly independent families. We merge the two
natural ways of generalizing independent families through a filter or an ideal
and we focus on the $\mathcal{C}$-independent families, where $\mathcal{C}$ is
the club filter. Also we show a relationship between the existence of
$\mathcal{J}$-independent families and the saturation of the ideal
$\mathcal{J}$.
