Canada's Prime Minister has resigned; how do they select the new leader?

The Liberal Leader in Canada has resigned today (06-Jan-2025), and halted parliament while a leadership search is conducted.

In Parliamentary systems, the Prime Minister is not elected, and so no new election is required to change the Prime Minister. The Prime Minister is whoever can command the confidence of the House of Commons. If one party has a majority, that will generally be the leader of that party. If no party has a majority, as is the current situation, then it’s subject to negotiation between parties to find a candidate who will have majority support. If no one can command the confidence of Parliament, that’s when a new election is needed.

There must always be a Prime Minister and a government, so Justin Trudeau will remain in a caretaker role until there is a replacement for him.

The Liberal Party won’t necessarily remain in power, if other parties who between them have a majority in the Commons can agree a coalition and a new Prime Minister. However, this seems unlikely.