Okay, I know the Femshep/Garrus marriage topic happened pages upon pages ago, but I thought I'd give my two cents.
The only remotely similar thing to "marriage" we have heard alluded to in Mass Effect with turians is that one guy who mentions having a mate. I'm suspicious that this is less a ceremonial or legal thing than it is just a mutual decision to stay together. It's also possible that turians go through a courtship process before settling on a mate for life--but bear in mind that even wild animals that mate for life can and do cheat on their mates. I don't believe asari normally marry either (due to lifespan most likely) and neither do krogan. Thus far, there's only one race other than humans that I think we've seen that definitely does marriage--the drell.
I've seen fanfics where people try to come up with some ritual or ceremony for Femshep to become mates with Garrus, or they've had him try to take up human culture and actually marry her--but until I have evidence supporting some form of turian marriage arrangement, I believe a turian makes friends with a female, eventually decides he wants to be more than friends, and if it works out they become mates by mutual agreement. The human perception would probably be that they decide to be lovers or cohabitants. In all honesty, this fits the general idea of turian society's personal freedoms--rather than having to go through a legal process of divorce, or shaking things up by going out and meeting strangers and getting obsessed, you stick with someone you already know you get along with, and if you drift apart, you can part ways without a lot of legal red tape (though I wouldn't idealize the drifting apart bit because I'm sure these things could still get ugly--something turian society would seriously frown on).
On the one hand, the term "mate" made me think it must be more unofficial, and yet also that it's possible turians actually mate for life (and poor turians if they end up with someone they don't like...). But, there are people in the real world who'll call their spouses their mates, so the vibe I got from that turian on Omega may be totally wrong (but interesting because it also implies he's either estranged from his mate, or doesn't want her in a place like Omega... or maybe both).
If turians do mate for life, then it makes me question the role of "blowing off steam" like Garrus did (sounded a lot like hate sex!). Sex as a form of courtship maybe? In which case, maybe it wasn't hate sex at all, but the situation somehow didn't work out. She was a recon scout, and they were heading into a dangerous mission. Maybe the reason it didn't come to more than that is because she died during the mission. But, turian personal freedoms imply that they're okay with casual sex, so on the other hand, they may also be perfectly okay with sleeping around and a "mate" may simply be the person a turian keeps coming back to or chooses to raise offspring with. Frankly, though, my suspicion is that a turian might have casual sex all he/she wants without a thought to how well he/she gets along with those partners, but would approach someone he/she considers a potential mate (in terms of mates for life) more respectfully--I guess it's more an idea of "stress relief/having fun" vs. "I want to spend the rest of my life with this person and maybe have babies together." If turians instinctively seek out a mate for life, then once again, being more attracted to friends than random strangers they find physically appealing would make sense--because it'd really suck to be stuck with someone it turned out you hated. I'm sure that would happen anyway, unless turians are less prone to changing over time than human beings are.
If on the other hand turians do not mate for life (though the term would generally imply they would) then maybe we should be looking at this more as a girlfriend/boyfriend thing, or even some kind of sociopolitical arrangement. I think in some ancient European cultures people would arrange to be handfasted for a year and a day or something. When the contract ended you could move on to someone else.
Among humans, marriage developed as a means of ensuring advantages to one or both families, and when you get down to it, most cultures did this by treating females as property--if a man had a daughter, he could sell her to another man for land or animals or other forms of wealth and improve his standing and his sons' inheritance. Of course it's also always (until recently) ultimately been about producing and raising children, which isn't necessarily always the case in modern society with the advent of various forms of birth control. Has one turian gender ever been considered unequal to the other? This is another question you have to ask when you're thinking about whether or not turians marry (and therefore whether FemShep can or would marry Garrus). If so, then marriage-style ceremonies are more likely. However, if turians always mated for life then it becomes less likely again because it's something that just happens--unless they have some kind of celebration or ceremony once they realize they've found that special someone, just to let everyone else know they're off-limits now.
I'll go out on a limb and say that the only reason Garrus would consider being with Femshep is that he is, in fact, a very bad turian. He's apparently pretty serious about being with her once she suggests the idea--likely only expressing doubt because he's not sure if she's serious about it. We don't know how serious Garrus is though--is he trying to woo her into becoming his mate? Choosing a human for a mate means he won't be contributing to the turian gene pool or to making any important sociopolitical connections with other turian families (I suspect that is a consideration for turians), and in fact, it will royally tick off many, many turians who remember the First Contact War/Relay 314 Incident and may cause discontent among the Cerberus crew members. It's a selfish decision. I suspect even asari aren't completely accepted as partners for turians, even though you see a lot of turian-asari pairs in-game. They'd be more acceptable than humans, though, due to the asari's long history with turians and the Council.