Some cases of Serre's uniformity problem

Given a number field $K$, Serre's uniformity question asks whether there exists a
constant $C_K$ (depending only on $K$) such that if $E/K$ is an elliptic curve
without complex multiplication, then $\bar{\rho}_{E,p}$ is surjective for every
prime $p>C_K$. In this talk, building on previous work by Darmon and Merel,
I will show that Serre's uniformity question has a positive answer for the family
of elliptic curves defined over the rationals, without complex multiplication and
admitting a non-trivial cyclic isogeny. I will then present some more recent
work I have been conducting on other families of elliptic curves.