Files Copy to the drive and disappear when put in New Computer

I use my External Hard Drive as a device for transferring files from one computer to another.  For a couple of months now I have found that when I copy and paste/ or drag and drop files into the hard drive it seems to work fine and the files are saved on the hard drive.  But then when I go to another computer to view those files and continue working on them they are not there anymore.  They just disappear into thin air.  They are on the drive one second and once I unplug it and plug it in to another computer they disappear.  No matter the other computer is. 

Is there any other way to transfer files where to make sure they are on the drive?  Please help, I am at a loss of what to do to fix this very frustrating problem!


Ok, so let me see if a get this right? If you copy files in your drive from computer A and then connect into Computer B you are not able to see the files?

Does the same situation happens with another storage device? Or backward? Let’s say from computer B to computer A? It happens with all the files?