After the French-Prussian war of 1870, the modern French Republic was established, which with a short interruption during World War 2 and some constitutional changes afterwards, still excists.
The third Republic was instituted after the collapse of the second French Empire as a result of the French-Prussian war. The third Republic excisted until the German invasion of France in May 1940. After this France was divided in Vichy-France, German occupied France, Italian occupied France and the Forces Françaises Libres. After the Second World War and the Provisional Government of the French Republic, the Fourth French Republic was established, excisting until 1958. Changes in the Republican system, formally made the Fifth French Republic from 1958 until today.