Cardinality estimation (CardEst) still remains a challenging problem for DBMS. Recent years have witnessed the success of ML-based cardinality estimators in outperforming traditional methods. However, these solutions suffer from poor generalizability to new data or query distribution, inability to handle complex queries, and substantial data preparation overhead, thus preventing their wide adoption in the real-world DBMS. Some recent efforts have been dedicated to addressing some but not all of these issues. We notice that the recent emerging Large Language Models (LLMs) have shown their remarkable generalizability to unseen tasks, capabilities to understand complex programs, and power to perform data-efficient fine-tuning. In light of this, we propose to leverage LLMs to mitigate the above issues. Specifically, we carefully craft prompts, and subsequently perform fine-tuning and self-correction during inference with LLMs for CardEst task. We then extensively evaluate LLMs' in-distribution and out-of-distribution generalizability, feasibility to support complex queries, and training data efficiency during fine-tuning LLMs on pre-training datasets. The results suggest that LLMs outperform the state-of-the-art in almost all settings, thus indicating their potential for the CardEst task. We further measure the end-to-end query execution time in DBMS by using the estimated cardinalities of LLMs in some practical settings, which suggests that the inference overhead of LLMs can be outweighed by the benefits brought by LLMs for CardEst.