Graph classification is a fundamental but challenging issue for numerous real-world applications. Despite recent great progress in image/video classification, convolutional neural networks (CNNs) cannot yet cater to graphs well because of graphical non-Euclidean topology. In this article, we propose a walk-steered convolutional (WSC) network to assemble the essential success of standard CNNs, as well as the powerful representation ability of random walk. Instead of deterministic neighbor searching used in previous graphical CNNs, we construct multiscale walk fields (a.k.a. local receptive fields) with random walk paths to depict subgraph structures and advocate graph scalability. To express the internal variations of a walk field, Gaussian mixture models are introduced to encode the principal components of walk paths therein. As an analogy to a standard convolution kernel on image, Gaussian models implicitly coordinate those unordered vertices/nodes and edges in a local receptive field after projecting to the gradient space of Gaussian parameters. We further stack graph coarsening upon Gaussian encoding by using dynamic clustering, such that high-level semantics of graph can be well learned like the conventional pooling on image. The experimental results on several public data sets demonstrate the superiority of our proposed WSC method over many state of the arts for graph classification.