  We introduce the concept of essential numerical range $W_{\!e}(T)$ for
unbounded Hilbert space operators $T$ and study its fundamental properties
including possible equivalent characterizations and perturbation results. Many
of the properties known for the bounded case do \emph{not} carry over to the
unbounded case, and new interesting phenomena arise which we illustrate by some
striking examples. A key feature of the essential numerical range $W_{\!e}(T)$
is that it captures spectral pollution in a unified and minimal way when
approximating $T$ by projection methods or domain truncation methods for PDEs.
