TITLE: Models for, and motivation for, (oo,n)-categories for general n
QUESTION [5 upvotes]: First: Is there a precise meaning to the term "model for (oo,n)-categories"? A related question, which might actually be the same question, is: what exactly do we want to get out of (oo,n)-category theory for general n? Whatever definition of (oo,n)-categories we use, what are the desired things it should satisfy? What should the main examples be, for general n? I know that bordism categories should be examples. What else? Actually, aside from the cobordism hypothesis and other TQFT-y things I don't really know what the motivations are for (oo,n)-category theory for general n (or at least n bigger than 1), so I hope that people can say some words about that as well. (For n=1, there seems to be a lot of motivation, see for example this or this or this or this or ...)
Second: Presently, what are the models that we have for (oo,n)-categories? Which models have been proven to be equivalent? Of course, there's already a lot about this on the nLab:
(oo,1)-categories
(oo,2)-categories
(oo,n)-categories
I'm mainly just curious about the current status on (oo,n)-categories for general n. Aside from n-fold complete Segal spaces, are there other definitions/"models"?

REPLY [4 votes]: The nlab page on n-categories includes a list of known definitions and comparisons, which should in particular include all definitions of (∞,n)-categories (since any definition of (∞,n)-category can be specialized to a definition of n-category = (n,n)-category by requiring all k-cells to be trivial for k>n).  This list is almost certainly incomplete, but I humbly propose that anyone with knowledge of something it omits should rectify the situation!  This sort of question is so common, and the comparisons are so important to the subject, that it would be useful to have a definitive list of what is known.