Logitech make bluetooth mice and keyboards but their proprietary stuff is better.
The devices are paired with the dongle, several devices can be paired with one dongle and vice versa. Once you've paired the devices (and if you buy a kit it's done for you) they behave like a wired keyboard and mouse.
Want to move from one computer to another, just unplug the dongle and stick it in the other computer. Doesn't matter what that device is as long as it has support for USB mouse/keyboard it will just work.
Bluetooth pairing depends on the quality of the bluetooth software on the computer. You have to place the device in a discoverable mode then get the computer to search then supply a pin and if you're lucky then it works but the next day it might be working or you might be back to square one.
Also my logitech mouse lasts 18 months on a single AA battery, they keyboard can go for 3 years. Bluetooth devices need charging once a week or new batteries once a month.
The concept of bluetooth is great but I've always found it to be a major let down where HID is concerned.