One of the big advances in symplectic topology in the 90s was Donaldson's theorem that when the symplectic class is integral, high multiples of its dual are represented by symplectic submanifolds.
These submanifolds behave like hyperplane sections in algebraic geometry; for instance, they satisfy the Lefschetz hyperplane theorem. They form the fibres of "symplectic Lefschetz pencils". Their intersections can be made to give symplectic submanifolds dual to multiples of wedge-powers of $\omega$.
Imagine first that $M$ is a compact complex manifold, $L\to M$ a hermitian, holomorphic line bundle, whose Chern connection has curvature $-2\pi i\omega$, a closed $(1,1)$-form. Then the zero-set of a $C^\infty$ section $s$ of $L^{\times k}$, if cut out transversely, is dual to $k[\omega]$. If $\omega$ is positive, the Kodaira embedding theorem then tells us that $L$ is ample: its high powers have enough holomorphic sections to embed $M$ into projective space. If $M$ is merely symplectic, with $-2\pi i\omega$ the curvature of some unitary connection in a hermitian line bundle, we can choose an almost complex structure $J$ on $M$ and consider transverse sections $s_k$ of $L^{\otimes k}$ for which, asymptotically, the $(0,1)$-part of $\nabla s_k$ along $s_k^{-1}(0)$ is much smaller than the $(1,0)$-part. Then $s_k^{-1}(0)$ will not quite be a $J$-holomorphic submanifold, but for $k \gg 0$ its tangent spaces will be close enough to being $J$-linear that it will still be a symplectic submanifold.
References: S. K. Donaldson, "Symplectic submanifolds and almost-complex geometry", J. Differential Geom. Volume 44, Number 4 (1996), 666-705; "Lefschetz pencils on symplectic manifolds", J. Differential Geom. Volume 53, Number 2 (1999), 205-236.
These papers are brilliant both geometrically and analytically: the analysis is mostly low-tech but extremely subtle.
