Large language models (LLMs) have recently become remarkably good at improving developer productivity for high-resource programming languages. These models use two kinds of data: large amounts of unlabeled code samples for pretraining and relatively smaller amounts of labeled code samples for fine-tuning or in-context learning. Unfortunately, many programming languages are low-resource, lacking labeled samples for most tasks and often even lacking unlabeled samples. Therefore, users of low-resource languages (e.g., legacy or new languages) miss out on the benefits of LLMs. Cross-lingual transfer learning uses data from a source language to improve model performance on a target language. It has been well-studied for natural languages, but has received little attention for programming languages. This paper reports extensive experiments on four tasks using a transformer-based LLM and 11 to 41 programming languages to explore the following questions. First, how well cross-lingual transfer works for a given task across different language pairs. Second, given a task and target language, how to best choose a source language. Third, the characteristics of a language pair that are predictive of transfer performance, and fourth, how that depends on the given task.