Svolgere is yet another verb starting with S, meaning there is likely a verb without the S, at its roots.
The use of the "prefix" S to give a word the opposite meaning is a common Italian phenomenon. It comes up frequently (see, for example this lesson). There is no fool-proof "rule," but knowing about the S-prefix can often give us a clue about a word. If we try a search of the word without the S, we might gain a deeper understanding of the word. Sometimes the S provides a different slant on a word, and isn't necessarily a negation or an opposite.
So if we look up volgere, we find that it does exist. We just don't use it very often in everyday conversation. Svolgere, on the other hand, is very common, but it's not easy to guess its meaning.
Let's take a closer look.
When the verb is in its non-reflexive form it can be translated as "to carry out," "to conduct," "to do," or "to perform." It's transitive. We use it a lot when the question is, "What does it do?" or "What do you do (as a job)?"
Ha una capacità di memoria elevatissima; può svolgere la stessa funzione di cinquemila calcolatori meccanici messi insieme, ma in un tempo infinitamente più breve.
It has a very high memory capacity; it can perform the same function as five thousand mechanical calculators put together, but in an infinitely shorter time.
Captions 3-5, Adriano Olivetti La forza di un sogno Ep.2 - Part 19Play Caption
Ci troviamo nel centro tartarughe WWF di Lampedusa, fa parte del progetto italiano del WWF, che svolge attività di conservazione sulle tartarughe marine,
We are at the WWF center in Lampedusa, it's part of the Italian WWF project, which conducts work on conserving sea turtles
Captions 36-38, WWF Italia Progetto tartarughe - Part 1Play Caption
Espressione del lavoro di ricerca che svolgono durante il loro soggiorno romano.
An expression of the research work they carry out during their stay in Rome.
Caption 10, Villa Medici L'arca della bellezza - Part 4Play Caption
When we use the reflexive form of the verb, we often translate it as "to take place." We could also say "to unfold" in certain contexts. The reflexive form is intransitive.
Una parte del film si svolge qua dove sembra veramente che il passato e il futuro siano coesistenti.
One part of the film takes place here where it really seems that the past and the future coexist.
Captions 34-35, Fratelli Taviani La passione e l'utopia - Part 6Play Caption
The reflexive form svolgersi, is extremely common, but not all that easy to guess at, since it's not a cognate... or is it?
If we look up the etymology of the verb svolgere, we do find volgere, but another, archaic, version of volgere — volvere, no longer in use, is mentioned as well. And if we try hard, we can see the verb "to evolve" as a sort of cognate. If we think of the verb svolgersi as something like, "to evolve," it might help us remember it.
How does this story evolve? Come si svolge questa storia?
If we look at the conjugation chart of the verb svolgere and we look at the conjugation chart of the verb svoltare (to change directions, to turn) there are some similarities, so this can be a bit confusing.
Both the non-reflexive and the reflexive form of the verb svolgere can mean "to unfold." So they intersect in a way. But we should just keep in mind that the non-reflexive form is transitive (it takes a direct object) and the reflexive form is intransitive (you won't find a direct object after it).
If you do a search of svolgere, and svolgersi on the Yabla videos page, you will have an overview of how these verbs are used. If you then go to the transcript for a given video where the word is used and hit command or control F to search the word there, you'll see the larger context, together with the English translation. You will see that the translation isn't consistent. Sometimes it's tricky to find the right word, since there really isn't a good, reliable English cognate.
Certainly, the two forms of svolgere are great verbs to have in your toolbox. If you pay attention, you will start hearing both of them a great deal. And now you know what they mean!