64-bit software on 32-bit hardware

Can anyone please tell me if 64-bit software will run on
32-bit hardware ?

I've googled for this but while there's plenty of information
on compatibility of 64-bit CPUs with 32-bit software, I
couldn't find any satisfactory treatment of the opposite case.

I understand that things are still in the development stage
and that it may be difficult to give a clear-cut answer, but
I'll be grateful for some indications.

The reason I'm asking is that I (and probably thousands of
others) want to buy a new computer now, but either do not
want to or cannot afford to invest in a 64-bit machine right
now. But many of us have to hang on to the same machine
(albeit with some upgrades) for at least a few years and
we wouldn't want to be stuck with something that will be
totally incapable of running 64-bit games and apps that
will be released in the foreseeable future.

I know a lot of people who are quite happily playing today's
games on their 4-year old Pentium 3s by adding more RAM
and newer graphics cards. Will the situation be quite different
when most new games and other software have adopted
the 64-bit platform ?