  Let $ X=(X_0,X_1)$ and $ Y=(Y_0,Y_1)$ be Banach couples and suppose $T: X\to
Y$ is a linear operator such that $T:X_0\to Y_0$ is compact. We consider the
question whether the operator $T:[X_0,X_1]_{\theta}\to [Y_0,Y_1]_{\theta}$ is
compact and show a positive answer under a variety of conditions. For example
it suffices that $X_0$ be a UMD-space or that $X_0$ is reflexive and there is a
Banach space so that $X_0=[W,X_1]_{\alpha}$ for some $0<\alpha<1.$
