Hidden Markov models are probabilistic graphical models based on hidden and observed random variables. They are popular to address classification tasks for time series applications such as part-of-speech tagging, image segmentation, genetic sequence analysis. We focus on direct extensions of these models, the pairwise and triplet Markov models. These models aim at relaxing the assumptions underlying the hidden Markov chain by extending the direct dependencies of the involved random variables or by considering the addition of a third latent process. While these extensions define interesting modeling capabilities that have been little explored so far, they also raise new problems such as defining the nature of their core probability distributions and their parameterization. Once the model is fixed, the unsupervised classification task (i.e. the estimation of the parameters and next of the hidden random variables) is a critical problem. We address these challenges in this paper. We first show that it is possible to embed recent deep neural networks in these models in order to exploit their full modeling power; we also consider a continuous latent process in triplet Markov chains which aims at estimating the nature of the joint distributions of the hidden and observed random variables, in addition to their parameters. For each model that we introduce, we propose an original Bayesian unsupervised estimation method which can take into account the interpretability of the hidden random variables in terms of signal processing classification. Through unsupervised classification problems on synthetic and real data, we show that the new models outperform hidden Markov chains and their classical extensions usually considered in the literature.