Reconstruction of PET images is an ill-posed inverse problem and often requires iterative algorithms to achieve good image quality for reliable clinical use in practice, at huge computational costs. In this paper, we consider the PET reconstruction a dense prediction problem where the large scale contextual information is essential, and propose a novel architecture of multi-scale fully convolutional neural networks (msfCNN) for fast PET image reconstruction. The proposed msfCNN gains large receptive fields with both memory and computational efficiency, by using a downscaling-upscaling structure and dilated convolutions. Instead of pooling and deconvolution, we propose to use the periodic shuffling operation from sub-pixel convolution and its inverse to scale the size of feature maps without losing resolution. Residual connections were added to improve training. We trained the proposed msfCNN model with simulated data, and applied it to clinical PET data acquired on a Siemens mMR scanner. The results from real oncological and neurodegenerative cases show that the proposed msfCNN-based reconstruction outperforms the iterative approaches in terms of computational time while achieving comparable image quality for quantification. The proposed msfCNN model can be applied to other dense prediction tasks, and fast msfCNN-based PET reconstruction could facilitate the potential use of molecular imaging in interventional/surgical procedures, where cancer surgery can particularly benefit.