Everywhere in Durham is maximum of 20 minutes walk away, so don't be put off by thinking something is a little off the beaten path.
Course (North Road) does a fine dining tasting menu, Barrio Comido does a fine dining Mexican one on a Saturday both about £60+booze.
My fav' is Lebaneat, food is perfect for me.
Loads of decent Indian restaurants (or as people have said to me many times "Indian owned, Bangladeshi ran and the food is cooked for English people") and starting to get decent Asian restaurants on North Road, near the centre & in Gilesgate that are ran by & cater for the growing Asian student population in Durham.