I'm shocked that Garrus-related search terms are STILL at the top of my blog's stats page. People, we need to slow down on the Garrus obsession!
...
Actually, no, we don't. HEE.
Anyway.
I noticed that a lot of people have been using the search phrase "how do I romance Garrus?" to find my blog, so I thought I'd provide some answers.
Long Story Short: Complete Garrus' Loyalty Quest. When you've gone on a number of missions in Mass Effect 2, Garrus will eventually want to talk to you to ask you for help in a personal matter. Agreeing to help him and completing the quest will make him Loyal to you, and make it possible for you to romance him.
If you need a longer explanation, click the "Continue reading" link below.
WARNING: THE SUCCEEDING PARAGRAPHS WILL CONTAIN SPOILERS.
I'll try to keep spoilers at a minimum, but I honestly can't see how I can avoid putting spoilers in a How to Romance Garrus guide.
AN ADDITIONAL WARNING BASED ON OTHER PLAYERS' OBSERVATIONS
As mentioned in the comments below, you might have trouble romancing Garrus if:
- You're romancing someone else in the Mass Effect 2 squad.
- You romanced Kaidan in the first Mass Effect game, kept him alive, and didn't tell Kelly it was over between the two of you.
Step One: Recruit Garrus
Obviously, you'll need to get Garrus into your squad so that you can romance him. The Illusive Man will provide three dossiers for people you can recruit into your squad. One of them is a dossier for "Archangel", who can be found on Omega.
EDI and Miranda will recommend that you go after "The Professor" first. That's actually a good way to go about it; you'll need "The Professor" to unlock the Tech Lab on the Normandy SR-2 to upgrade your weapons/armor/ship. If you're as crazy about our favorite turian and as suicidal as I am, you can go after Archangel first - upgrades be damned.
Once you rescue Archangel from the mercs that are ganging up on him, Garrus will join your squad.
Step Two: Complete his Loyalty Quest
After a few missions, your squad members will start asking you for help on personal matters. These are their Loyalty Quests. Completing these quests will secure their loyalty to you and open up romance options. I'd tell you when Garrus' Loyalty Quest activates, but I forgot exactly how many missions it took before it showed up. 
His Loyalty Quest will involve tracking down and attempting to assassinate someone. Once you've helped Garrus find the target, you're going to have to choose to either let Garrus go through with the assassination or convince him not to take the shot. Whatever you choose to do, you'll complete the quest and earn his loyalty.
(It's also interesting to note that Garrus reacts differently to what your Fem!Shep says to him during the Loyalty Quest depending on what you did with Dr. Saleon in Mass Effect.)
Step Three: Flirt with him!
After you finish his Loyalty Quest, Garrus will tell you an entertaining story about an experience he had while preparing for a high-risk mission on a turian ship. I'm sure all of you can figure out which among the dialogue options are the flirty ones without me spelling it out for you. :p
If you're done with this step, congratulations! Garrus is now your love interest.
Step Four: Stay interested in him!
Garrus will eventually have another conversation with your Fem!Shep, and all you have to do here is to reassure him that you like him. After that dialogue, you will have the option to break the romance off whenever you want by telling him that you just "want to be friends", but why ever would you want to do that?
Step Five: ENJOY.
Right before the final suicide mission, a certain turian will come to visit your Fem!Shep at her cabin. I'm sure you can take it from there. 
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Ack, so I did what you said, but Garrus…he turned me down!
Something about not wanting to be a name on my list of conquests…
I had been sexing up Jacob quite a lot in the game prior to being able to talk to Garrus, and I did have a thing with Kaiden in the first game. I’m guessing my pure sex driveness towards Jacob had something to do with this, even though I hadn’t gotten to the point where I had to break it off with him, because I did Jacob’s loyalty mission quite a bit after Garrus’. Know anything for certain?
Whoa, I’m not sure what went down there – I played as a Shepard that romanced Jacob first, but I didn’t have any issues with Garrus when I decided to romance him instead (aside from Jacob getting all miffed and calling Garrus a cuttlebone).
I’ve never encountered any complaints about either of my two my Shepards “lists of conquests” from Garrus…and neither of them have romanced Kaidan. That makes me think you were right in assuming that that probably had something to do with it.
Hi I think the Kaiden romance drove garrus away. At some point in the 2nd game you’re talking to someone that asks how you feel about kaiden after seeing him again on horizon. I think it’s kelly. You should say that you don’t feel for him or say it’s history now. That will probably make Garrus available to you. Just don’t flirt with Kelly too much because it becomes an option at that point
. ( As for a source…. I romanced Kaiden in one and was able to romance garrus in two no prob by doing this)
I have a glitch too. I haven’t really been flirting with anybody and I’ve progressed Garrus as far as I can. He said he’d do me he just needs to do some research. Mordin isn’t talking about sex so I don’t know what the deal is.
OJSKNCKSAIFDHIJAKJNCAICH. I’m so in love with him! x3 Thanks for this, I looked at guides and when I saw some of the stuff they said about it, I thought I royally screwed my profile over.
Hey, no problem! It’s always fun to help other fangirls out. Garrus is the best (even if I think ME3 sucks)! :p