Do those born Jewish welcome Jewish converts?

If I were to convert to Judaism and met a Jewish man, would their family be likely to approve of me? I've always had a fondness of the Jewish culture, in fact out of all the religions in the world it's the only one that makes me feel safe and I feel extremely drawn to it. I don't want to convert just to marry a Jewish man, but it'd be nice if I did meet a Jewish man to know that I'd be accepted, even though I wasn't born Jewish.