I've began to study the relationship between calibrations and holonomy, mainly through D.D. Joyce's Riemannian Holonomy Groups and Calibrated Geometry and partly through internet material.
Pretty much everyone explains this relationship by the holonomy principle: if $H=\text{Hol}_x$ and $\varphi_0$ is an $H$-invariant $k$-form in $T_pM$, then there is a parallel $k$-form $\varphi$ in $M$ with $\nabla\varphi=0$. In particular, this means $d\varphi=0$. Rescaling $\varphi_0$ if necessary, we get that $\varphi$ is a calibration.
So far, so good.
After this, people start saying something about special holonomy and invariably mention Berger's classification.
1) What does special mean in this context? I thought this was an informal adjective used by Joyce, but apparently everyone uses it and I haven't found a definition for it.
2) I understand Berger's list is interesting, for they deal with irreducible manifolds. But why don't they mention symmetric manifolds, which are not on the list, like $\mathbb{R}^n$, $\mathbb{S}^n,\mathbb{R}H^n$, compact Lie groups etc. They seem pretty interesting (and numerous) to me, so why not consider them?
(1) A generic metric has (restricted) holonomy group $SO(n)$ (More proper statement: the set of holonomy $SO(n)$ metrics is comeagre in the space of all Riemannian metrics). Hence the adjective special is coined (as in the opposite of "generic") when we can reduce it to smaller subgroups. It definitely predates Joyce (certainly Harvey and Lawson used that in their seminal paper introducing calibrations in the early 1980s).
(2) Elie Cartan proved that for Riemannian symmetric spaces $G/H$, the restricted holonomy group is the identity component of the isotropy group $H$. So this is just a pure algebra problem as to which Lie group is a subgroup of another Lie group (or equivalently which Lie algebra is a subalgebra of another), hence not interesting (as in having little if not no geometry content).