Purpose To develop an improved k-space reconstruction method using scan-specific deep learning that is trained on autocalibration signal (ACS) data. Theory Robust Artificial-neural-networks for k-space Interpolation (RAKI) reconstruction trains convolutional neural networks on ACS data. This enables non-linear estimation of missing k-space lines from acquired k-space data with improved noise resilience, as opposed to conventional linear k-space interpolation-based methods, such as GRAPPA, which are based on linear convolutional kernels. Methods The training algorithm is implemented using a mean square error loss function over the target points in the ACS region, using a gradient descent algorithm. The neural network contains three layers of convolutional operators, with two of these including non-linear activation functions. The noise performance and reconstruction quality of the RAKI method was compared with GRAPPA in phantom, as well as in neurological and cardiac in vivo datasets. Results Phantom imaging shows that the proposed RAKI method outperforms GRAPPA at high (≥4) acceleration rates, both visually and quantitatively. Quantitative cardiac imaging shows improved noise resilience at high acceleration rates (rate 4: 23% and rate 5: 48%) over GRAPPA. The same trend of improved noise resilience is also observed in high-resolution brain imaging at high acceleration rates. Conclusion The RAKI method offers a training database-free deep learning approach for MRI reconstruction, with the potential to improve many existing reconstruction approaches, and is compatible with conventional data acquisition protocols.