Title: LEXA: Language-agnostic Cross-consistency Training for Question Answering Tasks. Abstract: Cross-lingual information retrieval (CLIR) is a knowledge-intensive NLP task that requires a lot of domain-specific data in different languages. In previous works, authors were mostly using machine translation and iterative training for data mining. We considered the problem from another angle and present a novel cross-lingual pre-training and fine-tuning approach for CLIR tasks based on cross-lingual alignment. We present a new model LEXA-LM significantly improving cross-lingual knowledge transfer thus achieving new state-of-the-art in cross-lingual and monolingual question answering and cross-lingual sentence retrieval. Moreover, we show that our pre-training technique LEXA is a very powerful tool for a zero-shot scenario allowing to outperform some supervised methods.  