One of the major difficulties in graph classification is the lack of mathematical structure in the space of graphs. The use of kernel machines allows us to overcome this fundamental limitation in an elegant manner by addressing the pattern recognition problem in an implicitly existing feature vector space instead of the original space of graphs. In this paper we propose three novel error-tolerant graph kernels -- a diffusion kernel, a convolution kernel, and a random walk kernel. The kernels are closely related to one of the most flexible graph matching methods, graph edit distance. Consequently, our kernels are applicable to virtually any kind of graph. They also show a high degree of robustness against various types of distortion. In an experimental evaluation involving the classification of line drawings, images, diatoms, fingerprints, and molecules, we demonstrate the superior performance of the proposed kernels in conjunction with support vector machines over a standard nearest-neighbor reference method and several other graph kernels including a standard random walk kernel.