One of the key tasks in the design of large-scale dialog systems is classification. This consists of assigning, out of a finite set, a specific category to each spoken utterance, based on the output of a speech recognizer. Classification in general is a standard machine-learning problem, but the objects to classify in this particular case are word lattices, or weighted automata, and not the fixed-size vectors for which learning algorithms were originally designed. This chapter presents a general kernel-based learning framework for the design of classification algorithms for weighted automata. It introduces a family of kernels, rational kernels, that combined with support vector machines form powerful techniques for spoken-dialog classification and other classification tasks in text and speech processing. It describes efficient algorithms for their computation and reports the results of their use in several difficult spoken-dialog classification tasks based on deployed systems. Our results show that rational kernels are easy to design and implement, and lead to substantial improvements of the classification accuracy. The chapter also provides some theoretical results helpful for the design of rational kernels.