A new method, providing a significant improvement in sensitivity, is presented for obtaining selective two-dimensional NMR correlation (COSY) spectra. The method is demonstrated on a small molecule having nonlinear optical properties and is further demonstrated by the precise determination of scalar coupling constants in the peptide gramicidin and in a 57 amino acid fragment of the nucleocapsid of the hepatitis C virus, In these systems we observe sensitivity improvements of between a factor of 2 and 6. In the case of the protein it was not practical to record the selective COSY spectra using conventional techniques. The method makes use of the recently introduced concept of `'linear'' selective pulses which provide a simple way of obtaining pure-phase band selective excitation with an arbitrary shape for the response. In a selective 2D COSY experiment the relative sensitivity improvement provided by these pulses is cubed with respect to a 1D experiment. The resulting pulse sequences are highly unusual in that they require a modulation of the pulse shape and pulse length as the indirect detection period is increased.