Research on (multi-domain) task-oriented dialog (TOD) has predominantly focused on the English language, primarily due to the shortage of robust TOD datasets in other languages, preventing the systematic investigation of cross-lingual transfer for this crucial NLP application area. In this work, we introduce Multi2WOZ, a new multilingual multi-domain TOD dataset, derived from the well-established English dataset MultiWOZ, that spans four typologically diverse languages: Chinese, German, Arabic, and Russian. In contrast to concurrent efforts, Multi2WOZ contains gold-standard dialogs in target languages that are directly comparable with development and test portions of the English dataset, enabling reliable and comparative estimates of cross-lingual transfer performance for TOD. This enables us to detect and explore crucial challenges for TOD cross-lingually. We then introduce a new framework for multilingual conversational specialization of pretrained language models (PrLMs) that aims to facilitate cross-lingual transfer for arbitrary downstream TOD tasks. Using such conversational PrLMs specialized for concrete target languages, we systematically benchmark a number of zero-shot and few-shot cross-lingual transfer approaches on two standard TOD tasks: Dialog State Tracking and Response Retrieval. Our experiments show that, in most setups, the best performance entails the combination of (i) conversational specialization in the target language and (ii) few-shot transfer for the concrete TOD task. Most importantly, we show that our conversational specialization in the target language allows for a much more sample-efficient few-shot transfer for downstream TOD tasks.