In this paper, we introduce Target-Aware Weighted Training (TAWT), a weighted training algorithm for cross-task learning based on minimizing a representation-based task distance between the source and target tasks. We show that TAWT is easy to implement, is computationally efficient, requires little hyperparameter tuning, and enjoys non-asymptotic learning-theoretic guarantees. The effectiveness of TAWT is corroborated through extensive experiments with BERT on four sequence tagging tasks in natural language processing (NLP), including part-of-speech (PoS) tagging, chunking, predicate detection, and named entity recognition (NER). As a byproduct, the proposed representation-based task distance allows one to reason in a theoretically principled way about several critical aspects of cross-task learning, such as the choice of the source data and the impact of fine-tuning.