  We introduce the family of Theta operators $\Theta_f$ indexed by symmetric
functions $f$ that allow us to conjecture a compositional refinement of the
Delta conjecture of Haglund, Remmel and Wilson for $\Delta_{e_{n-k-1}}'e_n$. We
show that the $4$-variable Catalan theorem of Zabrocki is precisely the
Schr\"{o}der case of our compositional Delta conjecture, and we show how to
relate this conjecture to the Dyck path algebra introduced by Carlsson and
Mellit, extending one of their results. Again using the Theta operators, we
conjecture a touching refinement of the generalized Delta conjecture for
$\Delta_{h_m}\Delta_{e_{n-k-1}}'e_n$, and prove the case $k=0$, extending the
shuffle theorem of Carlsson and Mellit to a generalized shuffle theorem for
$\Delta_{h_m}\nabla e_n$. Moreover we show how this implies the case $k=0$ of
our generalized Delta square conjecture for
$\frac{[n-k]_t}{[n]_t}\Delta_{h_m}\Delta_{e_{n-k}}\omega(p_n)$, extending the
square theorem of Sergel to a generalized square theorem for
$\Delta_{h_m}\nabla \omega(p_n)$. Still the Theta operators will provide a
conjectural formula for the Frobenius characteristic of super-diagonal
coinvariants with two sets of Grassmanian variables, extending the one of
Zabrocki for the case with one set of such variables. We propose a
combinatorial interpretation of this last formula at $q=1$, leaving open the
problem of finding a dinv statistic that gives the whole symmetric function.
