We introduce the concept of the extension spectrum of a Hilbert space operator. This is a natural subset of the spectrum which plays an essential role in dealing with certain extension properties of operators. We prove that it has spectral-like properties and satisfies a holomorphic version of the Spectral Mapping Theorem. We establish structural theorems for algebraic extensions of triangular operators which use the extension spectrum in a natural way. The extension spectrum has some properties in common with the Kato spectrum, and in the final section we show how they are different and we examine their inclusion relationships.