Not 'ludus', which though itself a Latin translation of Greek 'schole' (originally 'leisure', then 'leisurely debate, philosophical stance, etc.') refers to a primary school.

There were many famous and important schools of higher learning at Rome and in many cities through the Empire from the middle of the 2nd century BCE on into the 5th century CE, and even later in certain outlying regions. These included institutions, some private and some sponsored by the government, where subjects including law, philosophy, medicine, and especially rhetoric comprised the standard curricula of post-secondary Roman education.