Any idea how to do an Autonumber?

Put the value '1' in a column in your data set.
In the cell beneath this value type =, click on
the cell above, type +1, and [Enter]. Copy this
formula down to the last row of your data.

Or if you want the numbers to adjust automatically when rows are deleted and for blank rows to be un-numbered try this one:
=IF(A1<>"",COUNTA($A$1:A1),"")

Or even simplier ...

1. Enter '1' in cell A1
2. Enter =OFFSET(A2,-1,)+1 in cell A2, and Copy
Down.

Or... if you have a column label in cell A1, then
use =SUM(OFFSET(A2,-1,),1) in cell A2 and below.
That'll allow you to even delete the 1st row of
data without any consequences!

But these last two formulas don't work if a row is inserted.

I does if you using a Data Form... If you're not
just use Ctrl-' (single quote) to fill in the
new, empty cell.

But this doesn't update the rest of the numbers, does it?

Uhhg, the Ctrl-' suggestion wasn't a good one.
Actually, what I like to do, but didn't want to
elaborate before is:

1. With cell A2 active, define a name (e.g, "counter")
with a reference of "=SUM(OFFSET(Sheet1!A2,-1,),1)".
2. Then use "=counter" in cell A2 and below.
3. ...then you can use Ctrl-' to reliably fill-in
a missing value.

You beat me to the punch. See my comment right
before yours.

Sorry to be such a nuisance, but what if you want to insert a blank row (and being blank, it should not have a number) ?

It was my understanding that the original poster
wanted to number all records; therefore, that
field should be filled in at all times as it would
if you were using the Data Form... menu command.