So Matt and Karen have done an interview explaining why they don't think romance is necessarily important for the new series of Doctor Who.
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All good points. The show is essentially about having mad adventures in time and space, and one of the best parts of that is the Doctor/Companion friendship- best mates having fun.
However.
I think it's safe to say that, though romance might not be necessary for the show, it has certainly played a large part in all of the seasons so far... and possibly even this current season. There have been a lot of episodes that emphasized romantic relationships, or at least the possibility of such relationships. The Doctor and Rose might be the obvious relationship to point out, since there was at least one onscreen admission of love that played a big part in the series... but that's certainly not the only example.
There have been a LOT of onscreen kisses, which were pretty much absent in the old series (from what I understand, at least.) The Doctor has kissed Rose, Reniette, Martha, Joan (okay, that was John Smith but still), Astrid, Donna, and Christina, and we know there is going to be an umcoming Doctor/Amy kiss from the previews. Guest characters are constantly asking if the Doctor and his companion are a couple, and even though they deny it, that's still been a recurring theme throught the show. And so far, all the main companions have had a story resolution in which they ended up with someone. Rose ended up with the Doctor (shut up, he IS the Doctor), Martha ended up with Mickey (:D) and Donna ended up with Shaun. There has been a lot of speculation that Amy will end up with Rory. And obviously romance is important to the fans, when you consider the tons of shippy fanfiction that has been written and the communities dedicated to certain relationships. Even had there been no onscreen indication of romantic feelings, fans would have inferred them anyway... I'm pretty certain this happened to the classic series.
So maybe romance isn't necessary to Doctor Who... but you can't deny the writers have a habit of inserting it anyway. And I, for one, have enjoyed it. ;)