I'm created a query from whatever the user passes through textboxes.

Sometimes these have apostrophes... As an example, O'KEEFE is entered
into the textbox...

This is part of the code
strSQL = Update ...(omited)... MasterTable.OWNER1_SURNAME = '" &
Replace(txtOWNER1_SURNAME, "'", "' & '")

Evaluates to:
MasterTable.OWNER1_SURNAME = 'O' & 'KEEFE'

How can I handle these apostrophes and account for null text boxes if
they exist?

I've also tried using:

IIf(txtOWNER1_SURNAME Is Not Null, (Replace(txtOWNER1_SURNAME, "'",
"''")), "")

I sorted it...

Just use:

& Chr$(34) & txtOWNER1_SURNAME &Chr$(34)

Then puts double qoutes around txtOWNER1_SURNAME (e.g. "O'BRIAN")