  We study loops which are universal (that is, isotopically invariant) with
respect to the property of flexibility ($xy\cdot x = x\cdot yx$). We also
weaken this to semi-universality, that is, loops in which every left and right
isotope is flexible, but not necessarily every isotope. One of our main results
is that universally flexible, inverse property loops are Moufang loops. On the
other hand, semi-universally flexible, inverse property loops are
diassociative. We also examine the relationship between universally flexible
loops and middle Bol loops. The paper concludes with some open problems.
