My (perhaps inaccurate) impression is there are many competing definitions of the notion of a (weak) $n$-category, none of which are generally accepted. I've run across several properties such definitions should satisfy, e.g. that weak $n$-groupoids should model homotopy $n$-types, or that the collection of $n$-categories should naturally be an $n+1$-category. But I haven't found anything like a comprehensive list of conjectural properties a "good" theory of weak $n$-categories should satisfy.

Is there such a list somewhere in the literature?

If not, I'd of course be thrilled with an answer containing such a list.