Historically the Legislative Assembly in British crown colonies superseded the (usually unelected) Legislative Council as the colonial legislature, as the colony gained more internal self-government, in some instances becoming the lower house.
In Australia the Legislative Assembly is the lower house of parliament in New South Wales, Victoria, and Western Australia, while in South Australia and Tasmania, the lower house is called the House of Assembly. It is the sole parliamentary chamber in Queensland, the Northern Territory and Australian Capital Territory.
In Canada the Legislative Assembly is the name of the single house of the legislature in all provinces and territories except Quebec, where it is called the National Assembly, and Newfoundland and Labrador, where it is called the House of Assembly.
Politicians elected to a Legislative Assembly are usually referred to as Members of the Legislative Assembly (MLAs).