Optical coherence tomography (OCT) images of the retina are a powerful tool for diagnosing and monitoring eye disease. However, they are plagued by speckle noise, which reduces image quality and reliability of assessment. This paper introduces a novel speckle reduction method inspired by the recent successes of deep learning in medical imaging. We present two versions of the network to reflect the needs and preferences of different end-users. Specifically, we train a convolution neural network to denoise cross-sections from OCT volumes of healthy eyes using either (1) mean-squared error, or (2) a generative adversarial network (GAN) with Wasserstein distance and perceptual similarity. We then interrogate the success of both methods with extensive quantitative and qualitative metrics on cross-sections from both healthy and glaucomatous eyes. The results show that the former approach provides state-of-the-art improvement in quantitative metrics such as PSNR and SSIM, and aids layer segmentation. However, the latter approach, which puts more weight on visual perception, outperformed for qualitative comparisons based on accuracy, clarity, and personal preference. Overall, our results demonstrate the effectiveness and efficiency of a deep learning approach to denoising OCT images, while maintaining subtle details in the images.