  In a recent paper by Cook, et al., which introduced the concept of unexpected
plane curves, the focus was on understanding the geometry of the curves
themselves. Here we expand the definition to hypersurfaces of any dimension
and, using constructions which appeal to algebra, geometry, representation
theory and computation, we obtain a coarse but complete classification of
unexpected hypersurfaces. In particular, we determine each $(n,d,m)$ for which
there is some finite set of points $Z\subset\mathbb P^n$ with an unexpected
hypersurface of degree $d$ in $\mathbb P^n$ having a general point $P$ of
multiplicity $m$. Our constructions also give new insight into the interesting
question of where to look for such $Z$. Recent work of Di Marca, Malara and
Oneto \cite{DMO} and of Bauer, Malara, Szemberg and Szpond \cite{BMSS} give new
results and examples in $\mathbb P^2$ and $\mathbb P^3$. We obtain our main
results using a new construction of unexpected hypersurfaces involving cones.
This method applies in $\mathbb P^n$ for $n \geq 3$ and gives a broad range of
examples, which we link to certain failures of the Weak Lefschetz Property. We
also give constructions using root systems, both in $\mathbb P^2$ and $\mathbb
P^n$ for $n \geq 3$. Finally, we explain an observation of \cite{BMSS}, showing
that the unexpected curves of \cite{CHMN} are in some sense dual to their
tangent cones at their singular point.
