It depends on the details.
1) you must have a SmartThings hub/dongle which can act as a “Matter Controller.” right now the choices are a SmartThings V2 hub, a SmartThings V3 hub, an Aeotec hub, the SmartThings Station, or the dongle that can be added to the Family Hub refrigerator. They continue to add more options to this list. You cannot use matter devices with only the smartthings app: you have to have a “Matter controller” hub/dongle as well.
2) you must have the SmartThings app on an iOS or android device to act as a “matter commissioner.“ At the time of this writing, the official web interface could not perform this function.
If you have a Samsung galaxy phone, it has some additional matter options that other choices do not have, but nothing particularly important.
3) If the device you want to add is individually certified for Matter over Wi-Fi such as the Leviton D215p smart plug, then that’s all you need as long as your Wi-Fi router either supports IPv6 addresses or can spoof them. (Matter-certified devices using Ethernet instead of WiFi also need to support IPv6 addresses.) Just make sure the device you want to add is on the same Wi-Fi/LAN network as the controller and commissioner devices in one and two above, and you should be able to add it, although the exact steps will depend on whether you have previously added it to a different matter controller or not.
Note: if you are in the United States, you use one of the major Internet providers, and your wifi router is less than four years old, you are probably ready for IPv6 addresses, although you may need to change your router setting. But if your Wi-Fi router is older than that, or you do not get your Internet service from one of the major US carriers, you may have more work to do to get ready before using any matter devices.
Google has a test page which will tell you if your network supports IPv6 or not. Make sure you visit the page from a device connected to the same Wi-Fi network that your matter devices will use.
The following is a good video explaining the issue.
https://www.youtube.com/watch?v=eqVAYkoOKOY
4) If the device you want to add is individually certified for matter over thread, such as the Eve Energy smart plug, you also need to have a Thread border router. The SmartThings V3 hub, the Aeotec hub, and the SmartThings Station are all thread border routers. The V2 hub is not. But there are a number of other thread border routers available, such as the Apple HomePod mini, some Google home smart speaker models, some echo models, etc. so you may already have one.
The following is a very good explainer article about thread border routers:
What is a Thread Border Router? | Know-how | matter-smarthome
5) many devices are not individually certified for matter, but can be brought into matter through a matter bridge. This includes many zigbee devices, if their brand (or a third-party company) offers a “Matter bridge“ that can share those nonMatter devices with other matter commissioners. This is similar to how a hue bridge has always worked without matter to expose its connected devices to other home automation platforms.
For example, the aqara M2 hub is also a Matter bridge. So you can add some specific Aqara zigbee Sensors and switches that are not individually certified for matter to that hub, then add that hub to smartthings, and it will bring in some of its other devices with it. (But not necessarily all devices or all features.)
So in this device class, the matter bridge is certified for matter, but the individual end devices are not. This is also similar to how bridge devices work with Apple HomeKit.
At the time of this writing, smartthings/Aeotec hubs do not work as a matter bridge, so you cannot add them to other platforms. And there are as yet no plans to add that feature.
But there are a few matter Bridges from other brands that can now be added to your smartthings account using matter as long as you have the devices mentioned in one and two above. These include the SwitchBot hub 2 and some of the aqara hubs. Tuya and IKEA have announced that they intend to have some hub models which are also matter bridges, but those have not yet been released. Philips Hue released its matter bridge support for the hue hub in September 2023.
So, if you have the SwitchBot Hub 2, you could add it to your SmartThings account using matter, and it would bring in any curtainbots that you had connected to it. This would then give you a local connection to the curtainbots, which is not available through the pre-existing cloud to cloud integration. But again, you have to have both the matter controller from one above and the matter commissioner from two above set up.
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So, as you can see, the answer to your question depends on the specific device you hope to add, and on the details of your smartthings set up.
On the smartthings side, you must have a SmartThings/Aeotec hub/dongle that can act as a Matter controller.
On the end device side, it depends whether that device uses matter over Wi-Fi, matter over thread, or a matter bridge.
If the end device is individually certified for matter over Wi-Fi, that’s all you need.
If the end Device is individually certified for Matter over thread, you also need at least one thread border router in your house.
If the end device is not individually certified for matter, but can connect to its own brand’s hub which can act as a “Matter Bridge“, then you can probably bring it into your SmartThings account that way.
You will not be able to add your smartthings/Aeotec hub to other matter platforms, because at the time of this writing, it only acts as a “matter controller,” it does not act as a “matter bridge.” 
if you have all of the requirements as listed above, but you are getting an error 39 – something when trying to add the device, make sure that both your hub and the device you are trying to add have the most current firmware.
For the smartthings/Aeotec hub, just unplug it, wait a few minutes, and plug it back in. That should cause it to re-sync with the smartthings cloud and get the newest firmware if it is available. For a V3/Aeotec hub, you need version 50 or above.
for the device you are trying to add, follow the manufacturer’s instructions to check the firmware level. For example, several community members reported different error 39 – somethings when they were trying to add a Tapo matter plug until they got the most recent firmware.
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Matter is a rapidly developing standard with a lot of changes, and it can be possible for things to get out of sync.
Research note: Logos and product descriptions on AliExpress are frequently inaccurate. Sellers often just copy pictures or take parts of product descriptions from other places without really understanding them. So you cannot rely on a Matter logo on a product listing on that site. Logos on sites for authorized dealers should be fine, as well as on the manufacturer’s own site or their official Amazon storefront.
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As of May 14, 2024, please do not ask questions about individual products displayed on AliExpress in this topic: it’s too much work for me to research those because the seller descriptions are so often inaccurate. Find the product description on the manufacturer’s own site or at an authorized reseller and you should be able to answer the question for yourself using the information in this post. If not, you can start a new topic requesting information about that specific seller listing, and hopefully other people will be able to help you.