We define the concept of a spectrally compact operator, and study the basic properties of these operators. We show that the class of spectrally compact operators is strictly contained in the class of compact operators and in the class of spectrally bounded operators. It is also proved that the set of spectrally compact operators on a spectrally normed space $E$ is a right ideal of $\mathrm{SB}(E)$ and in certain cases it is a two sided ideal. We will also study the spectral adjoint of a spectrally compact operator.