Most theoretical frameworks that focus on data errors and inconsistencies follow logic-based reasoning. Yet, practical data cleaning tools need to incorporate statistical reasoning to be effective in real-world data cleaning tasks. Motivated by these empirical successes, we propose a formal framework for unclean databases, where two types of statistical knowledge are incorporated: The first represents a belief of how intended (clean) data is generated, and the second represents a belief of how noise is introduced in the actual observed database instance. To capture this noisy channel model, we introduce the concept of a Probabilistic Unclean Database (PUD), a triple that consists of a probabilistic database that we call the intention, a probabilistic data transformator that we call the realization and captures how noise is introduced, and a dirty observed database instance that we call the observation. We define three computational problems in the PUD framework: cleaning (infer the most probable clean instance given a PUD), probabilistic query answering (compute the probability of an answer tuple over the unclean observed instance), and learning (estimate the most likely intention and realization models of a PUD given a collection of training data). We illustrate the PUD framework on concrete representations of the intention and realization, show that they generalize traditional concepts of repairs such as cardinality and value repairs, draw connection to consistent query answering, and prove tractability results. We further show that parameters can be learned in practical instantiations, and in fact, prove that under certain conditions we can learn a PUD directly from a single dirty database instance without any need for clean examples.