In this study, we measure the moral reasoning ability of LLMs using the Defining Issues Test - a psychometric instrument developed for measuring the moral development stage of a person according to the Kohlberg's Cognitive Moral Development Model. DIT uses moral dilemmas followed by a set of ethical considerations that the respondent has to judge for importance in resolving the dilemma, and then rank-order them by importance. A moral development stage score of the respondent is then computed based on the relevance rating and ranking. Our study shows that early LLMs such as GPT-3 exhibit a moral reasoning ability no better than that of a random baseline, while ChatGPT, Llama2-Chat, PaLM-2 and GPT-4 show significantly better performance on this task, comparable to adult humans. GPT-4, in fact, has the highest post-conventional moral reasoning score, equivalent to that of typical graduate school students. However, we also observe that the models do not perform consistently across all dilemmas, pointing to important gaps in their understanding and reasoning abilities.