Current and future imaging surveys will measure cosmic shear with a statistical precision that demands a deeper understanding of potential systematic biases in galaxy shape measurements than has been achieved to date. We investigate the effects of using the point spread function (PSF) measured with stars to determine the shape of a galaxy that has a different spectral energy distribution (SED) than the star. We demonstrate that a wavelength dependent PSF size, for example as may originate from atmospheric seeing or the diffraction limit of the primary aperture, can introduce significant shape measurement biases. This analysis shows that even small wavelength dependencies in the PSF may introduce biases, and hence that achieving the ultimate precision for weak leasing from current and future imaging surveys will require a detailed understanding of the wavelength dependence of the PSF from all sources, including the CCD sensors.