What is the analogue of the principal eigenvalue for elliptic operators with non-compact resolvents? Focusing on the case where the lack of compactness is due to the unboundedness of the domain, we show that the answer depends on the property one is looking for: existence of a positive eigenfunction, simplicity, lower bound of the spectrum, characterization of the maximum principle. Indeed, there is not a unique notion fulfilling all such properties in general. In the last part of the talk we present some recent results concerning degenerate elliptic operators.