Multilingual task-oriented dialogue (TOD) facilitates access to services and information for many (communities of) speakers. Nevertheless, the potential of this technology is not fully realised, as current datasets for multilingual TOD-both for modular and end-toend modelling-suffer from severe limitations. 1) When created from scratch, they are usually small in scale and fail to cover many possible dialogue flows. 2) Translation-based TOD datasets might lack naturalness and cultural specificity in the target language. In this work, to tackle these limitations we propose a novel outline-based annotation process for multilingual TOD datasets, where domainspecific abstract schemata of dialogue are mapped into natural language outlines. These in turn guide the target language annotators in writing a dialogue by providing instructions about each turn's intents and slots. Through this process we annotate a new large-scale dataset for training and evaluation of multilingual and cross-lingual TOD systems. Our Cross-lingual Outline-based Dialogue dataset (termed COD) enables natural language understanding, dialogue state tracking, and end-toend dialogue modelling and evaluation in 4 diverse languages: Arabic, Indonesian, Russian, and Kiswahili. Qualitative and quantitative analyses of COD versus an equivalent translation-based dataset demonstrate improvements in data quality, unlocked by the outline-based approach. Finally, we benchmark a series of state-of-the-art systems for cross-lingual TOD, setting reference scores for future work and demonstrating that COD prevents over-inflated performance, typically met with prior translation-based TOD datasets.