In a recent segment of Meraviglie, Alberto Angela uses a verb that looks familiar: sistemare. It must have something to do with "system," right?

The noun il sistema certainly exists, and is a true cognate of "the system" in English.

E allora con un ingegnoso sistema di raccolta delle acque,

And so with an ingenious system for collecting water,

riuscì a riempire ben sette cisterne che sono sparsi [sparse] per tutto il territorio.

he managed to fill a good seven cisterns that are scattered around the whole area.

Captions 36-37, In giro per l'Italia - Asciano - S. Giuliano Terme: Villa Bosniascki

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A detail to remember is that although it has a typically feminine ending, sistema is a masculine noun. In English, too, “system” has any number of connotations.

So the noun sistema is fairly straightforward, but English doesn't really have a corresponding verb to go with sistemare. Sistemare might even fall into the category of untranslatable Italian verbs, although it's an easy-to-figure-out untranslatable verb. Sistemare is a general, catch-all type of verb that can mean any number of things, depending on the context. 

When Alberto Angela tells us the fascinating story of a huge underground cistern in the city of Matera, what does he mean by sistemare? Good question.

Quando si è sistemata la piazza nel millenovecentonovantuno…

When the piazza was renovated in nineteen twenty-one…

Caption 12, Meraviglie - Ep. 1 - Part 15

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We see from the translation that the piazza was renovated, and we get this from the context of the documentary itself. But sistemare could also have referred to it being  "neatened up," "cleaned up," "put in order," "put to rights."

When you want to fix something up, make improvements, put things right, make minor repairs, put things in a certain place, make preparations, or even get your pet ready for the night, sistemare is a good verb.

In the following examples from Yabla videos, sistemare is used to mean "to work out," "to set up," and "to fix up."

Note that in the first example, the reflexive form sistemarsi is used.

Mi dispiace molto, Marika, e spero che tutto si sistemerà al più presto.

I'm really sorry, Marika. And I hope everything will work out as soon as possible.

Caption 41, Italiano commerciale - Difficoltà con colleghi e contratti

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Valter arrivava sempre prima per sistemare l'attrezzatura per gli allievi.

Valter always came early to set up the equipment for the students.

Caption 52, Il Commissario Manara - S2EP5 - Mondo sommerso

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Adesso hai quest'impressione perché lo vedi così tutto in disordine,

Now you have that impression because you're seeing it all messy,

quando sarà sistemato vedrai...

when it's fixed up, you'll see...

Captions 35-36, Un medico in famiglia - S1 EP1 - Casa nuova

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One general way of thinking about the verb sistemare is with "to take care of". 

You took care of an unpaid bill? L'hai sistemato. You took care of it.
Your plumber fixed that leaky faucet? L'ha sistemato. He took care of it. He fixed it.
You wrote a draft of an article? Lo devi ancora sistemare. You still have to fine-tune it.

We can also turn sistemare into a noun: una sistemata. In English, we might use a gerund for this, as in the first example below. 

You don't really want to give your kitchen a thorough cleaning at the moment, but you want it to look nice. Ci dai una sistemata (you give it a neatening up).
You ask your hairdresser, Mi dai una sistemata ai capelli (Will you give me a little trim)?

With the noun sistemata, we often use the verb dare (to give), which can also be used reflexively.

Dopo il viaggio, mi sono data una sistemata prima di presentarmi agli suoceri (after the trip I freshened up before meeting my in-laws/I gave myself a freshening up).

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Practice:

As you go through your day, as you take care of one problem after another, try using sistemare when you have succeeded, or when you haven't yet. Maybe you will even have fun taking care of these problems!

L'ho sistemato! Menomale. (I took care of that. Whew!)

Questo lo devo sistemare (I have to take care of this).

Ask someone else to help you take care of something — something that needs fixing, or a situation that needs resolving.

Me lo puoi sistemare (can you take care of this for me)?