Yuki and Machi's relation is the one that touched my heart from the bottom, the one that became special for me.
There are importan moments of their story that I'll never forget, because they're recounted with intensity so hot that I always think of them with a smile.

The problems with Machi's mother, the difficult relation between Yuki and Sohma Family, feelings of love for someone that can't requite you and the sespair to be useless in this world.
Certainly, it's only thanks to their meeting if they were able to change their lives, and, aobviously, to find the right person to love.

Fruits Basket is a series rich of great characters, and everyone has a story and a problem that the major part of the time, it's resolved by Tohru.
Yuki was in love with her since the begin of the series, and sincerely, I thought that nobody else could been able to reach his heart. Machi is different than all the others, she never tried to obtain the attention of someone, she was close in herself. And she lived alone, in a world in which Manabe was the only one ables to speak with her (but he never touch the deep of her heart).
Yuki has been able to reach Machi's heart and to release her lonely soul from the desperation.
Machi's tears were chatartic, and from the moment in which Yuki and her became closer, she changed piece by piece, finding a reason to smile and a person that will always be there for her, in every moment of her life.

I remember that reading Furuba they were the first ones that make me think that even without Tohru, the world continue to go on. I like Takaya-sensei's way to describe emotions and feelings of her characters, but Yuki and Machi are different than the others, their story is the sweetest and the most adorable one I've ever read from her.
I think of a lot of girls that, like Machi, are afraid to love because of the horrible things happened in their life, and reading her story and the development of her relation with Yuki, they can find hope, and a reason to think that one day, if they'll try to do their best, surely someone will catch their heart, saving them from loneliness.
An important message like this is the thing I like the most from Yuki and Machi's relation.

I think that if love exists, the one of Machi and Yuki it is. Because it's not at the first sight and not found in a simple episode dedicated to them, its' something beautiful that grown up with them and the development of the story.
A person ables to change another one in something best, surely is the one that will always have an important place in the heart and in the life of the other person, and thinking that Yuki and Machi mean this for me, I feel sure that love exists for all, we just have to wait the right person ables to open our heart.