Lists and dictionaries can be used to store and organize data in Python. Lists are mutable sequences that can be modified after they are created. Common list operations include accessing elements by index or slice, checking membership with "in", concatenating lists, and modifying elements. Dictionaries store data as key-value pairs and allow looking up values using almost any type of key, making them useful for storing related data like databases. Common dictionary operations include accessing values with keys and methods like get, keys, items, update and clear.