I attended a conference once where the invited speaker, Raul Rojas, showed that the Z3 was a universal computer, which wasn't previously believed to be the case. The basic idea of the proof was to show that you could have a single while-loop on the Z3 containing a big case-statement in the body of the loop which basically simulates a (finite) Turing machine.

The interesting part of this talk however was this: the Z3 did support neither loops nor conditional branches per se! Instead, it could just compute a series of mathematical operations encoded on a punched tape.

To simulate the case statement without the conditional branching, the trick you do is you execute every branch of the case-statement each time, but you only allow one of the branches to write the results of its computation back to the machine's memory.

But how do you get a while-loop? Well, you just glue the ends of the punch tape together! ;-)

(More details here: http://www.zib.de/zuse/Inhalt/Kommentare/Html/0684/universal...)

Also, from wikipedia:

"The Z1 was a mechanical computer designed by Konrad Zuse from 1935 to 1936 and built by him from 1936 to 1938."

and

"The Atanasoff–Berry Computer (ABC) was...conceived in 1937, the machine was not programmable, being designed only to solve systems of linear equations. It was successfully tested in 1942."

I think I would argue that the first computer in the sense we accept now was the worlds first stored program computer - the Manchester SSEM:

http://en.wikipedia.org/wiki/Small-Scale_Experimental_Machin...

MIT press has an excellent book, "The First Computers- History and Architectures"[1] that discusses a number of potential "firsts" based on varying criteria- well worth a read, if you're into this sort of thing. I'd agree that the Z1 was the first complete, working machine that I consider a computer.

[1]http://www.amazon.com/First-Computers--History-Architectures...

Interesting, I was unaware that it was shown there was an electronic, general purpose computer that predated the ENIAC.