The reason for this article is because a lot of us don't physically see our hard drives they are often remote in a datacenter etc and the actual serial number we see in SMART is not enough to check for some manufacturers.
A good example is our first one the Toshiba
=== START OF INFORMATION SECTION ===
Device Model: TOSHIBA DT01ACA200
Serial Number: 33FMDW4AS........
Here's the link to the warranty checker and from there you can also create an RMA:
Note that the serial# shown in SMART is not complete as is the case for many other brand sof HDDs.
For example a 2TB Toshiba shows the following Serial Number in SMART: