A foundational goal of linguistics has been to understand why languages look the way they do. A range of possible explanations exist – from domain-specific representations to cognition-external factors like history and grammaticalization – and all of these undoubtedly play some role. But determining exactly how these link with specific features of language remains challenging, and the role of domain-specific mechanisms has been particularly contentious. In this paper, I highlight a growing new approach, which uses artificial language experiments to link individual-level biases to cross-linguistic trends in language structure. Using word and morpheme order as case studies, I will show how a range of different paradigms and learner populations allow us to make progress on this crucial issue in linguistics. I will focus on typological trends in word and morpheme order. For some ordering trends, experimental evidence points to variation across populations, suggesting that the best explanation for these patterns likely has its root in language history and grammaticalization. In other cases, the evidence points to the role of universal but domain-general cognitive biases, like transparency and simplicity. These domain-general biases interact with linguistic representations in important ways. Taken together these studies help adjudicate between alternative explanations for a number of specific ordering patterns and suggest a new sense of domain-specificity in the evolution of language.