An individualised head-related transfer function (HRTF) is very important for creating realistic virtual reality (VR) and augmented reality (AR) environments. However, acoustically measuring high-quality HRTFs requires expensive equipment and an acoustic lab setting. To overcome these limitations and to make this measurement more efficient HRTF upsampling has been exploited in the past where a high-resolution HRTF is created from a low-resolution one. This paper demonstrates how generative adversarial networks (GANs) can be applied to HRTF upsampling. We propose a novel approach that transforms the HRTF data for direct use with a convolutional super-resolution generative adversarial network (SRGAN). This new approach is benchmarked against three baselines: barycentric upsampling, spherical harmonic (SH) upsampling and an HRTF selection approach. Experimental results show that the proposed method outperforms all three baselines in terms of log-spectral distortion (LSD) and localisation performance using perceptual models when the input HRTF is sparse (less than 20 measured positions).