It's one of the most controversial issues of the New Who series. Should the Doctor be written in a romantic sense? Should his companions be allowed to fall in love with him? Could he ever fall in love in return? The fans are hotly divided on this issue.
In Old Who it wasn't an issue because the BBC had very strict rules on the subject. There were actually written guidelines for how the Doctor could and could not touch his companions, to avoid even a hint of anything happening behind the scenes (i.e. the bedroom).
However when the made for TV movie appeared, it featured a very prominent romantic subplot. So when it came time to revive the show in 2005, there was a whole new range of possibilities for the writers to play around with.
The question is... should they?
[SPOILER WARNING -- If you have not yet watched ALL 6 seasons of New Who, I can guarantee there will be spoilers.]
[...I did warn you about spoilers...]
[Still here? Okay, read at your own risk.]
To Romance or not to Romance, that is the question.
There are a lot of fans, probably not a majority, but a very vocal minority, who believe that the Doctor should absolutely always be asexual. These tend to be those who were introduced to Old Who first, and therefore have the idea of a non-romantically-inclined Doctor firmly in their heads. I understand this. If I'd watched a show for almost thirty years (or seen a good chunk of that) without any romance involving the title character, I'd be pretty thrown for a loop if he suddenly started snogging people.
However, what has really kept the show alive over the years is its ability to change and adapt to the times. I mean, the most iconic feature of the show is that the Doctor regenerates any time they need a new actor in the part. It's a show that is not afraid to change, and that change has lead it through 48 years of intense fan love.
On the other hand, should the show become entirely focused on Romance? I don't think that's a good direction either. After all if you change something too much, you lose the core essence that has drawn people in for too long. So some change is good, but not too much.
But some fans do think any romance is too much. Is it really? Is the Doctor really that incapable of romantic love and sexual desire?
Absolutely. Kind of hard to end up with a granddaughter if you aren't into those things. Let's not forget that the Doctor's very first companion was his granddaughter Susan, and later on he mentions that he once was married and a father.
(Yes, I know, the novels do posit the idea that Time Lords are grown to adulthood on looms... but this isn't official canon, and anyhow, the show itself shows the Doctor and the Master as small children on Gallifrey, so I think that kind of debunks that myth)
Russell T Davis, the head writer who oversaw the relaunch in 2005 was obviously in the romance camp. He created the young girl, Rose, who became the Doctor's companion and eventually fall in love with him.
Her first Doctor was Christopher Eccleston, who played the doctor with his traditional quirkyness. And more importantly, he was rather homely to look at, which helped keep him in the 'eccentric professor' category instead of becoming a sex icon.
Then David Tennant appeared and quickly became the most popular Doctor ever according to fan polling, besting out even Tom Baker. This is certainly largely because of his huge talent and charisma... but it's undeniable that his appealing looks were also part of the picture.
And right away the romantic element of the show started amping up. "The Girl in the Fireplace" by Steven Moffatt was the first Doctor-centered love story on the show. Coming close on the heels of "School Reunion" which featured much tension between Rose and Sarah Jane Smith (who was now revealed to have been in love with the Doctor), this solidified the Doctor's position as a dashing romantic (I here mean romantic in the classic sense, ironically).
Then we got the very climatic and intensely romantic ending of season 2, followed by season 3 which thematically centered very strongly around Martha's unrequited love for the Doctor. He is now a recognized desirable hero and does Martha ever know it!
After such a romance laden third season, everyone was ready for a return to a more classic approach. With Donna as his companion, the Doctor was once again able to fall more into the eccentric professor role, and surprisingly enough, the platonic chemistry between David Tennant and Catherine Tate resulted in the strongest Doctor/Companion bond of 10's run.
Not that season four lacked romance. Steven Moffat once again took the lead on the romantic front and introduced us to the mysterious River Song, a woman who seems in every way the Doctor's equal... from his future. Is she his wife? His friend? Something else entirely? Viewers went slightly insane trying to figure it out (See my Doctor Who archive in the Geek Portal for my own crazed ideas).
And of course the grand finale resolved Rose's love story, but creating a Doctor duplicate, a move that even RTD himself admitted might not have been the best idea. However it satisfied a lot of fans and properly closed that door.
So the Doctor is off to a clean start when he regenerates into Matt Smith.
There are fans who find Matt Smith hot. Kudos to you. I'm not saying a word against that. He's a fantastic actor. And as the Doctor I find him perfectly odd, eccentric and fun. Really not romantic. Which is really what the show needed at this point.
Romance didn't disappear, far from it. If anything it became more prominent with Amy and Rory, an engaged couple... with Amy having a thing for the Doctor. Sort of. For awhile. Actually after season 6 the whole "Amy in Love with the Doctor" thing doesn't seem nearly as prevalent to me. I mean, now that we know she's his mother-in-law and all. ;)
In season 6 we got the full scoop on River Song. And we also got what I think is really the perfect romance for the Doctor. The Doctor remained his odd, eccentric, not traditionally attractive self... and yet he fell in love and a woman fell in love with him in what I felt was a very believable and appropriate way. They became equals, and while their story is incredibly romantic, it didn't make the Doctor a 'romantic hero.'
(Okay, he and River have insane chemistry and their scenes are really incredibly hot, but they manage this without taking away from the Doctor's 'eccentric old professor' persona which I think is amazing. Alex Kingston and Matt Smith for the win.)
So I'd say there have been some solid wins and also some misses on the romantic front. I believe romance can work in Doctor Who, but I also think it's been used a bit too much in the New Series. I think, however, now that the Doctor is married to River, the dynamtic is going to change and that will be a good thing.