  We deal with the complex issue of whether large cardinals are intrinsically
justified principles of set theory (we call this the Intrinsicness Issue). In
order to do this, we review, in a systematic fashion, (1.) the abstract
principles that have been formulated to motivate them, as well as (2.) their
mathematical expressions, and assess the justifiability of both on the grounds
of the (iterative) concept of set. A parallel, but closely linked, issue is
whether there exist mathematical principles able to yield all known large
cardinals (we call this the Universality Issue), and we also test principles
for their responses to this issue. Finally, we discuss the first author's
Structural Reflection Principles (SRPs), and their response to Intrinsicness
and Universality. We conclude the paper with some considerations on the global
justifiability of SRPs, and on alternative construals of the concept of set
also potentially able to intrinsically justify large cardinals.
