  The laminar flow past rectangular prisms is studied in the space of
length-to-height ratio ($1 \le L/H \le 5$), width-to-height ratio ($1.2 \le W/H
\le 5$) and Reynolds number ($Re \lessapprox 700$). The primary bifurcation is
investigated with linear stability analysis. For large $W/L$ it consists of an
oscillating mode breaking the top/bottom planar symmetry. For smaller $W/L$ the
flow becomes first unstable to stationary perturbations, and the wake
experiences a static deflection, vertical for intermediate $W/L$ and horizontal
for small $W/L$. Weakly nonlinear analysis and nonlinear direct numerical
simulations are used for $L/H = 5$ and larger $Re$. For $W/H = 1.2$ and $2.25$,
after the primary bifurcation the flow recovers the top/bottom planar symmetry
but loses the left/right one, via supercritical and subcritical pitchfork
bifurcations, respectively. Further increasing $Re$, the flow becomes unsteady
and oscillates around either the deflected (small $W/H$) or the non-deflected
(intermediate $W/H$) wake. For intermediate $W/H$ and $R$e, a periodic and
fully symmetric regime is detected, with hairpin vortices shed from the top and
bottom leading-edge (LE) shear layers; its triggering mechanism is discussed.
At large $Re$ and for all $W/H$, the flow approaches a chaotic state
characterised by the superposition of different modes: shedding of hairpin
vortices from the LE shear layers, and wake flapping in the horizontal and
vertical directions. In some portions of the parameter space the different
modes synchronise, giving rise to periodic regimes also at relatively large
$Re$.
