Assuming they do not cast defensively, when a spellcaster takes longer than 1 action to complete a spell (as is the case with long casting time spells and, more likely to see in combat, Summon Monster, Summon Nature's Ally, Antilife Shell, etc.) when is an Attack of Opportunity provoked? When they begin casting, during the entire round, when they end casting, when they begin or end casting, or something else entirely?
You provoke an attack of opportunity only once, at the beginning of the spell's casting.
From the section "Actions in Combat" in Chapter 8 of the Core Rulebook, under full round actions:
You only provoke attacks of opportunity when you begin casting a spell, even though you might continue casting for at least 1 full round.