We propose a statistical approach to speech-to-speech translation that uses finite-state models in all levels. Acoustic hidden Markov models (HMMs) model the pronunciation of the input-language phonemes and words, while the input-output word mapping, along with the syntax of the output language, are jointly modeled by means a large stochastic finite-state transducer. This allows for a complete integration of all the models so that the translation process can be performed by searching for an optimal path of states through the integrated network. As in speech recognition, HMMs can be trained from an input-language speech corpus, and the translation model is learned automatically from a parallel (text) training corpus. This approach has been assessed in the framework of the EuTrans project, funded by the European Union. Extensive experiments have been carried out with speech-input translations from Spanish to English and from Italian to English in applications involving the interaction (by telephone) of a customer with the front desk of a hotel. A summary of the most relevant results is presented.