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Accurate segmentation of kidneys and kidney tumors is an essential step for radiomic analysis as well as developing advanced surgical planning techniques. In clinical analysis, the segmentation is currently performed by clinicians from the visual inspection images gathered through a computed tomography (CT) scan. This process is laborious and its success significantly depends on previous experience. Moreover, the uncertainty in the tumor location and heterogeneity of scans across patients increases the error rate. To tackle this issue, computer-aided segmentation based on deep learning techniques have become increasingly popular. We present a multi-scale supervised 3D U-Net, MSS U-Net, to automatically segment kidneys and kidney tumors from CT images. Our architecture combines deep supervision with exponential logarithmic loss to increase the 3D U-Net training efficiency. Furthermore, we introduce a connected-component based post processing method to enhance the performance of the overall process. This architecture shows superior performance compared to state-of-the-art works using data from KiTS19 public dataset, with the Dice coefficient of kidney and tumor up to 0.969 and 0.805 respectively. The segmentation techniques introduced in this paper have been tested in the KiTS19 challenge with its corresponding dataset.