Text-based personality computing (TPC) has gained many research interests in NLP. In this paper, we describe 15 challenges that we consider deserving the attention of the NLP research community. These challenges are organized by the following topics: personality taxonomies, measurement quality, datasets, performance evaluation, modelling choices, as well as ethics and fairness. When addressing each challenge, not only do we combine perspectives from both NLP and social sciences, but also offer concrete suggestions. We hope to inspire more valid and reliable TPC research.