2-Plectic geometry is the higher generalization of symplectic geometry, the special case of n-plectic geometry (multisymplectic geometry) for $n = 2$. This is the input for higher prequantum geometry in degree 2.
As symplectic geometry naturally describes classical mechanics and, via geometric quantization, quantum mechanics, hence 1-dimensional quantum field theory, so 2-plectic geometry naturally describes 2-dimensional classical field theory and, via its higher geometric quantization, 2-dimensional QFT.
A semisimple Lie group is canonically a 2-plectic manifold, with the canonical 3-form $\langle -, [-,-]\rangle$ on the Lie algebra (the canonical Lie algebra cocycle), extended to a left invariant differential form.
A G2-manifold is manifold of dimension 7 characterized by carrying a 2-plectic form.
A (pre-)symplectic groupoid is a Lie groupoid equipped with a (pre-)2-plectic structure.
higher and integrated Kostant-Souriau extensions:
(∞-group extension of ∞-group of bisections of higher Atiyah groupoid for $\mathbb{G}$-principal ∞-connection)
(extension are listed for sufficiently connected $X$)
∞-Chern-Simons theory from binary and non-degenerate invariant polynomial
(adapted from Ševera 00)
Over smooth manifolds, the general setup is discussed in
and considered in the general context of higher differential geometry/extended prequantum field theory in
Domenico Fiorenza, Chris Rogers, Urs Schreiber,
Higher geometric prequantum theory,
L-∞ algebras of local observables from higher prequantum bundles
Applications to the 2-dimensional string sigma-model (see at string Lie 2-algebra) are discussed in
John Baez, Chris Rogers, Categorified Symplectic Geometry and the String Lie 2-Algebra, Homology Homotopy Appl. Volume 12, Number 1 (2010), 221-236. (arXiv:0901.4721,euclid).
John Baez, Alexander Hoffnung, Chris Rogers, Categorified Symplectic Geometry and the Classical String (arXiv:0808.0246)
A survey of some (potential) applications of 2-plectic geometry in string theory and M2-brane models is in section 2 of
and in