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 MULTI 2 WOZ, a new multilingual multidomain TOD dataset, derived from the wellestablished English dataset MULTIWOZ, that spans four typologically diverse languages: Chinese, German, Arabic, and Russian. In contrast to concurrent efforts (Ding et al., 2021;Zuo et al., 2021), MULTI 2 WOZ contains goldstandard 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. We then introduce a new framework for multilingual conversational specialization of pretrained language models (PrLMs) that aims to facilitate crosslingual 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 an exceptionally sample-efficient fewshot transfer for downstream TOD tasks.