Until a few weeks ago I was an architect at a bank; they have 2x IBM 390 mainframes, and they will have them for a very long time to come.
Reason is the applications they run are core banking applications which were written over a 15-20 year period using Natural (that's a programming language) and Adabas (a database that uses the organisational concept of Files, rather than Tables).
They have such a massive investment in these applications that it is going to take them a very long time to move to something else. I'm not just talking code and data, but big teams of mainframe people that 'know the business'.
Given that these are core banking systems and have been very reliable, they have little appetite to move to emulation, or another platform.
Drawbacks of mainframes today are primarily exorbitant support and maintenance fees, and a dwindling skills base in the market.
They also still run VMS, so that's kind of a minicomputer.
Hope that helps.