First thing is to be clear what a res publica is - no it is to complicated. What it is not is a government of the people through elected representatives, though elections played a role.

Cicero, through the mouth of Scipio Aemilianus, in de Republica, defines a good republic like this (2.56):

The Senate maintained the state in the following condition: though the people were free, not much was done through them; most things were done on the authority of the Senate according to custom and precedent. The consuls had a power which, though limited in length to one year, was in its nature and legitimacy equivalent to that of a king. And (a factor that was perhaps the most vital in maintaining the power of the aristocracy) the rule was staunchly retained whereby the people's corporate decisions were not valid unless endorsed by the Senate's authority... Yet, with the people's consent, all the power was firmly held in the hands of the leading citizens; and in those times great military feats were accomplished by valiant men who wielded supreme power, whether as dictators or consuls.

This isn't the sort of Republic you might be used to from modern political science textbooks. Now, Cicero was an odd duck, and I wouldn't take his word on Roman political theory as doctrine. He got himself exiled after all at least in part for his ideas about Roman rights and traditions.

But let's take this as a starting point. The key element is the primacy of the aristocracy through the institution of the Senate, and tradition by which popular will defers to the judgment of the Senate, and absolute authority of the magistrates. Cicero glosses this a little later as

the magistrates having adequate power, the aristocratic council adequate influence, and the people adequate freedom

(Polybius, who had actually taught Scipio Aemilianus, had a completely different idea of how a res publica functioned, but just leave his crazy Greek ideas aside for the time being)

Alright, when did this theoretical arrangement end? A modern Roman history textbook will put it somewhere around Augustus. Some might say 49, when Caesar crossed the Rubicon (which, there is a good argument to make, didn't really mean anything), some might say 44 when Caesar was assassinated, some 31 with the battle of Actium, 29, 27, somewhere around there.

All those dates have something to do with Julius Caesar's civil war, or Augustus' triumph over his opponents for leadership of the Julian party after Caesar's civil war. I suspect, at least in the Anglophone world, this is largely due to the influence of Syme, who in turn was influenced by Tacitus, who famously began his Annals a claim to Augustus establishing totalitarian rule:

When after the destruction of Brutus and Cassius there was no longer any army of the Commonwealth, when Pompeius was crushed in Sicily, and when, with Lepidus pushed aside and Antonius slain, even the Julian faction had only Caesar left to lead it, then, dropping the title of triumvir, and giving out that he was a Consul, and was satisfied with a tribune's authority for the protection of the people, Augustus won over the soldiers with gifts, the populace with cheap corn, and all men with the sweets of repose, and so grew greater by degrees, while he concentrated in himself the functions of the Senate, the magistrates, and the laws.

That's not the only choice possible though. Cicero as early as 54 (I think that's right?) was complaining that the Republic was dead. Velleius Paterculus, living under Tiberius (unless you want to call him a shill, which I think is unfair) thought he was living in a republic. And Augustus also claimed to have restored the republic after a period of civil war.

In recent scholarship, I think the last thing is Flower, Roman Republics, who proposed 6 different republics between the expulsion of the kings and Augustus. For her, “republic” is traditional and customary government of the aristocracy, roughly, with readjustments made so often. A strong break occurs with Sulla, who replaces tradition and custom with formalization and law, and following Sulla's dictatorship, she sees a period of anarchy or rule by strong men, until Augustus established monarchy.

I (since I was invited, and since Flower is already moving the “end of the Republic” around) have my own idea. I would put it in 70 with the passage of the lex de imperio Vespasiani, and my reasoning is this:

The one truly innovative thing Augustus did was accept the tribunician power. Private armies had be used before. Consecutive consulships had been held before. Age restrictions had been broken before. In my understanding (there is a Cicero quote to support me but I couldn't locate it – if anyone knows it ( /u/xenophontheathenian maybe?) I'd appreciate it) offices are the reward the people give to men who deserve from their service to the state to receive them. This is connected with dignitas and auctoritas, two different forms of political authority. This didn't change with Augustus – he relied strongly on his auctoritas as a mechanism for getting his people in the positions he wanted them. These are also traits that can be inherited to some extent, and the Julio-Claudian emperors had laws passed that recognized in a formal way their possession of these informal qualities. What Vespasian did with his lex was by law grant him those qualities – following Flower's model though with a different end point, I think this is where tradition and custom come to be replaced by law.

But as far as I know I'm the only person in the world that thinks that.