  Domain wall propagation in cylindrical nanowires with modulations of diameter
is a key phenomenon to design physics-oriented devices, or a disruptive
three-dimensional magnetic memory. This chapter presents a combination of
analytical modelling and micromagnetic simulations, with the aim to present a
comprehensive panorama of the physics of pinning of domain walls at
modulations, when moved under the stimulus of a magnetic field or a
spin-polarized current. For the sake of considering simple physics, we consider
diameters of a few tens of nanometers at most, and accordingly domain walls of
transverse type. Modeling with suitable approximations provides simple scaling
laws, while simulations are more accurate, refining the results and defining
the range of validity of the models. While pinning increases with the relative
change of diameter, a key feature is the much larger efficiency of pinning at
an increase of diameter upon considering current rather than field, due to the
drastic decrease of current density related to the increase of diameter.
