Predicting the first visibility of the young lunar crescent from a given location is an astronomical problem which has challenged astronomers and mathematicians for more than four millennia. Already in the second half of the first millennium BCE Babylonian priest-astronomers developed sophisticated numerical algorithms for predicting the motion of the moon, the times of its synodic phases and the first visibility of the lunar crescent above the western horizon just after sunset. More accurate methods were devised in later ages by Hindu, Islamic and Jewish scholars and the refinement of these methods has continued up until the present day.