  A theoretical study is presented on the scattering of graphene surface
plasmons by defects in the graphene sheet they propagate in. These defects can
be either natural (as domain boundaries, ripples and cracks, among others) or
induced by an external gate. The scattering is shown to be governed by an
integral equation, derived from a plane wave expansion of the fields, which in
general must be solved numerically but it provides useful analytical results
for small defects. Two main cases are considered: smooth variations of the
graphene conductivity (characterized by a Gaussian conductivity profile) and
sharp variations (represented by islands with different conductivity). In
general, reflection largely dominates over radiation out of the graphene sheet.
However, in the case of sharply defined conductivity islands there are some
values of island size and frequency where the reflectance vanishes and,
correspondingly, the radiation out of plane is the main scattering process. For
smooth defects, the reflectance spectra present a single maximum at the
condition $k_p a \approx \sqrt{2}$, where $k_p$ is the GSP wavevector and $a$
the spatial width of the defect. In contrast, the reflectance spectra of sharp
defects present periodic oscillations with period $k_p' a$, where $k_p'$ is the
GSP wavelength inside the defect. Finally, the case of cracks (gaps in the
graphene conductivity) is considered, showing that the reflectance is
practically unity for gap widths larger than one tenth of the GSP wavelength.
