General image super-resolution techniques have difficulties in recovering detailed face structures when applying to low resolution face images. Recent deep learning based methods tailored for face images have achieved improved performance by jointly trained with additional task such as face parsing and landmark prediction. However, multi-task learning requires extra manually labeled data. Besides, most of the existing works can only generate relatively low resolution face images (e.g., $128\times128$), and their applications are therefore limited. In this paper, we introduce a novel SPatial Attention Residual Network (SPARNet) built on our newly proposed Face Attention Units (FAUs) for face super-resolution. Specifically, we introduce a spatial attention mechanism to the vanilla residual blocks. This enables the convolutional layers to adaptively bootstrap features related to the key face structures and pay less attention to those less feature-rich regions. This makes the training more effective and efficient as the key face structures only account for a very small portion of the face image. Visualization of the attention maps shows that our spatial attention network can capture the key face structures well even for very low resolution faces (e.g., $16\times16$). Quantitative comparisons on various kinds of metrics (including PSNR, SSIM, identity similarity, and landmark detection) demonstrate the superiority of our method over current state-of-the-arts. We further extend SPARNet with multi-scale discriminators, named as SPARNetHD, to produce high resolution results (i.e., $512\times512$). We show that SPARNetHD trained with synthetic data cannot only produce high quality and high resolution outputs for synthetically degraded face images, but also show good generalization ability to real world low quality face images.