This talk provides an introduction to challenging statistical problems arising in the study of celestial objects: planets, stars, galaxies and the Universe as a whole. We start with a review of the close historical links between astronomy and statistics, from the ancient Greeks through Laplace and Gauss. However, the communities diverged during the 20th century, developing into a poor state with great needs for advanced methodology but weak links between the fields. This is ameliorating today with a vibrant subfield of astrostatistics. The talk then illustrates the diversity of astronomical datasets and outlines some of the methodological problems now under investigation. Challenges include treatments of: heteroscedastic measurement errors with known variances; non-standard censoring and truncation; irregularly-spaced time series; and model selection of exoplanets orbiting other stars. We end with opportunities for statisticians to become involved in these problems.