  We consider three-dimensional Quantum Electrodynamics in the presence of a
Chern-Simons term at level $k$ and $N_f$ flavors, in the limit of large $N_f$
and $k$ with $k/N_f$ fixed. We consider either bosonic or fermionic matter
fields, with and without quartic terms at criticality: the resulting theories
are critical and tricritical bosonic QED$_3$, Gross-Neveu and fermionic
QED$_3$. For all such theories we compute the effective potentials and the
$\beta$ functions of classically marginal couplings, at the leading order in
the large $N_f$ limit and to all orders in $k/N_f$ and in the couplings. We
determine the RG fixed points and discuss the quantum stability of the
corresponding vacua. While critical bosonic and fermionic QED$_3$ are always
stable CFTs, we find that tricritical bosonic and Gross-Neveu QED$_3$ exist as
stable CFTs only for specific values of $k/N_f$. Finally, we discuss the phase
diagrams of these theories as a function of their relevant deformations.
