Denoising has proven to be useful to efficiently generate high-quality Monte Carlo renderings. Traditional pixel-based denoisers exploit summary statistics of a pixel's sample distributions, which discards much of the samples' information and limits their denoising power. On the other hand, sample-based techniques tend to be slow and have difficulties handling general transport scenarios. We present the first convolutional network that can learn to denoise Monte Carlo renderings directly from the samples. Learning the mapping between samples and images creates new challenges for the network architecture design: the order of the samples is arbitrary, and they should be treated in a permutation invariant manner. To address these challenges, we develop a novel kernel-predicting architecture that splats individual samples onto nearby pixels. Splatting is a natural solution to situations such as motion blur, depth-of-field and many light transport paths, where it is easier to predict which pixels a sample contributes to, rather than a gather approach that needs to figure out, for each pixel, which samples (or nearby pixels) are relevant. Compared to previous state-of-the-art methods, ours is robust to the severe noise of low-sample count images (e.g. 8 samples per pixel) and yields higher-quality results both visually and numerically. Our approach retains the generality and efficiency of pixel-space methods while enjoying the expressiveness and accuracy of the more complex sample-based approaches.