  Flat surfaces that correspond to $k$-differentials on compact Riemann
surfaces are of finite area provided there is no pole of order $k$ or higher.
We denote by \textit{flat surfaces with poles of higher order} those surfaces
with flat structures defined by a $k$-differential with at least one pole of
order at least $k$. Flat surfaces with poles of higher order have different
geometrical and dynamical properties than usual flat surfaces of finite area.
In particular, they can have a finite number of saddle connections. We give
lower and upper bounds for the number of saddle connections and related
quantities. In the case $k=1\ or\ 2$, we provide a combinatorial
characterization of the strata for which there can be an infinite number of
saddle connections.
