BRUSSELS — The European Union is one of the most complex political experiments in history, with nearly half a billion voters in 28 member countries, glued together with thousands of laws, regulations and processes. That work is done largely through four institutions: the European Commission, the European Council, the European Parliament and the European Central Bank.

Getting the top job at one of these institutions, tasked with making policy out of cacophony in the bloc, is a singularly political process. On Thursday and Friday, the heads of the 28 European Union member states are convening in Brussels to make a first stab at the selection, a twice-a-decade exercise that is never smooth.

While guided by some principles, the process is ultimately deeply political, with heads of government promoting candidates who share their nationality or ideology for the coveted roles.

They will very likely need to meet again before filling the leadership posts. But what do all these presidents do?