The work presented here proposes a new voice conversion (VC) approach based on hidden Markov models (HMMs) for spectral conversion and excitation estimation. This paper is divided in two main parts: First, an initial HMM-based VC system is presented and compared to a state-of-the-art ML-GMM VC system in a monolingual conversion scenario with parallel training data; The second part shows the necessary modifications to use the HMM VC system in a cross-lingual conversion scenario and compares it with a cross-lingual VC system based on artificial neural networks (ANNs). The results of the tests show improved performance of the proposed HMM VC system compared with both the ML-GMM and ANN-based VC alternatives, while at the same time keeping most of the flexibility afforded by the ANN approach with respect to training data requirements.