I can install the game, no problem, but when I try to launch it, well nothing happens.
I tried using "run as administrator" as well and still the game refuses to launch. I re-installed the game many times, and it did nothing.
I've contacted the support team three times about this since september 3rd and I got absolutely no answer. No help. I'm counting on you guys...I would really like to be able to play a game I paid for.
Thanks.

You've been contacting GOG for almost two weeks and they haven't answered? Frankly, that's really surprising. Except for weekends, the GOG guys have always been been great about getting back to me within 24-48 hours.
Is the game failing after clicking the "Launch Shogo" button in the game launcher or is the game launcher not even running?

Even after clicking the launch shogo button it does nothing. The game is perfectly installed, but it refuses to launch, no matter what I do. I've even tried playing around with the compatibility settings. Nothing.
And still no answer from gog!

So you are getting to the launch app, but have you gone into any of the settings to try adjusting anything? At the very least, you should click on the Display button to set resolution, but you should also click on Advanced to check out those settings as well. I recommend at least enabling triple buffering.
Wow, incredibly strange that GOG hasn't responded. I wish I had an explanation or suggestion for that.

No. I am not getting to the launch app. After I finish installing the game, a GoG window appears to tell me that the game was installed and if I want to launch the game now, or just click on finish or something like that.
The game launch app never appears, I can't even get to the settings.
The compatibility stuff is set by right clicking on the icon and going to properties.
The game just doesn't respond to anything, as if it is not even installed!

Ah, the regular game launcher has a button labeled "Launch Shogo", hence my confusion.
Have you tried going to the game's install directory and running the executable directly by double-clicking on it? Does "Shogo.exe" even show up in the task manager when you try running it? Are you running any anti-virus, firewall or other security software that does active application "blocking" in any way?
Still no word from GOG support, huh?

I just wanted to say that I am having the exact same issue. In my case, I have noticed via Task Manager that Shogo.exe is running, but it doesn't appear on-screen, and there is no tab in the taskbar for it. I have not tried contacting GOG's support yet. I have purchased several games from GOG, and so far this is the only one to not work on my Vista machine. I *am* suspecting that McAfee might be blocking it but haven't checked that yet. (I need to do that.)

Update: I do not believe that McAfee is blocking the game. I need to check for malware though.