  We study the Cauchy problem on the real line for the nonlocal Fisher-KPP
equation in one spatial dimension, \[ u_t = D u_{xx} + u(1-\phi*u), \] where
$\phi*u$ is a spatial convolution with the top hat kernel, $\phi(y) \equiv
H\left(\frac{1}{4}-y^2\right)$.
  After showing that the problem is globally well-posed, we demonstrate that
positive, spatially-periodic solutions bifurcate from the spatially-uniform
steady state solution $u=1$ as the diffusivity, $D$, decreases through
$\Delta_1 \approx 0.00297$. We explicitly construct these spatially-periodic
solutions as uniformly-valid asymptotic approximations for $D \ll 1$, over one
wavelength, via the method of matched asymptotic expansions. These consist, at
leading order, of regularly-spaced, compactly-supported regions with width of
$O(1)$ where $u=O(1)$, separated by regions where $u$ is exponentially small at
leading order as $D \to 0^+$.
  From numerical solutions, we find that for $D \geq \Delta_1$, permanent form
travelling waves, with minimum wavespeed, $2 \sqrt{D}$, are generated, whilst
for $0 < D < \Delta_1$, the wavefronts generated separate the regions where
$u=0$ from a region where a steady periodic solution is created. The structure
of these transitional travelling waves is examined in some detail.
