We introduce the numerical spectrum $$\sigma _n(A)\subseteq {\mathbb {C}}$$ σ n ( A ) ⊆ C of an (unbounded) linear operator A on a Banach space X and study its properties. Our definition is closely related to the numerical range W(A) of A and always yields a superset of W(A). In the case of bounded operators on Hilbert spaces, the two notions coincide. However, unlike the numerical range, $$\sigma _n(A)$$ σ n ( A ) is always closed, convex and contains the spectrum of A. In the paper we strongly emphasise the connection of our approach to the theory of $$C_0$$ C 0 -semigroups.