I'm a geek, I have a special interest for Ancient Rome and I play computer games. Go figure. So here are my favorites, in several categories.
The ever-popular Caesar series (1 through 4) by Vivendi Universal, where you become a provincial governor within the empire and fulfill several missions, build your cities, from roads up, defend it against barbarians and deal with plagues, fires and other nuisances.
All in all, a nice game, though in the 4th part of the series I found the graphics too demanding (they're nice, but not that nice to justify the technical requirements) and the gameplay too easy. In the first part, I got stuck for weeks trying to pass a level, with the fourth, I was done completely before the week was over. Or maybe my geekiness level increased drastically in the meantime.
If you're into strategy games, Rome: Total War by Sega is simply the best. It's a combination of real time and turn based strategy; the real time battles are in 3D, with a great engine and beautiful graphics, that allow the display of hundreds of characters at the same time. The outcome of each battle greatly depends on your ability to maneuver your troops, which is quite unique, as far as I could find, and adds to the realism (we all know the winner of ancient battles wasn't always the one with the biggest army).
Instead of being just one character, you are the head of one of three major Roman families, each with unique features, and, as such, you can control all male members of your family, adopt new ones, issue orders for the commanders of your army and so on. Enjoyable, challenging, and not annoying at all from the historic point of view – give this game a try if you have the chance.
Yes, I've played my fair share of dash games. I know, lame.
Roads of Rome from Realore Studios is one such fine example; you are the brave warrior Victorius who has to build the Roman road network system before marrying a certain Julia. Granted, not the smartest game in town, and doesn't make even the vaguest attempt at historical accuracy, but fun, if you have an hour to kill.
Oh, talk about time killers, dash games don't even begin to compare to their match 3 counterparts. This could cost you a few sleepless nights, as you're just trying to gather a few hundred more resources to pass to the next level. Cradle of Rome (1 and 2) was one of the big hits in its category, and for the good reason: extremely simple gameplay (hey, it's match 3. It' easier than Solitaire), nice graphics, amusing minigames, just everything you might expect.
I wish I knew of a better Roman-themed game in this category, but I don't... so until I find something better, here it is: Romance of Rome, also from Realore Studios. Good game, as far as hidden objects games go these days, but, from my point of view, disappointing graphics. I was expecting something more... Roman.
There. Go play.