Did you know that LongWord in Delphi is not (any longer) always a 32 bit unsigned integer?
I turned out that the majority of the participants use Cardinal, followed by Uint32.
Only less than 10% use LongWord.
Stefan Glienke was the first to point out that LongWord is not always 32 bits, even though the 64-bit Windows Data Types Compared to 32-bit Windows Data Types entry in the DokWiki seems to suggest it. But it only covers Data types for Windows.
There is a better overview table on the Simple Types reference page. Even though I find it rather odd that they say that Integer and Int64 are aliases for NativeInt (I wouldn’t have had that “alias” column in the Platform Dependent Integer Types table at all.)
So, where does that leave me? I guess it’s best to switch to Int / UInt + bitness for those types where it is important how many bits an integer has (such as in records that define data written to a file). For the other cases it might be better to use NativeInt or NativeUInt, which translates to Int32 / UInt32 on 32 bit platforms and Int64 and UInt64 on 64 bit platforms. Unless of course they don’t.