Abstract: We begin by surveying what is known about the geometry and topology of the symplectic analogue of Calabi-Yau manifolds. 4-dimensional symplectic CYs resemble the Kähler CY surfaces topologically, in particular, their Betti numbers are bounded. Geometric analysis is essential in getting such bounds via Taubes' almost Kähler Seiberg-Witten theory. A main question is whether all symplectic forms on the K3 surface are Kähler forms, for which Donaldson suggested an approach via the almost Kähler CY equation. In contrast, in higher dimensions, symplectic CYs are known to be much more flexible, for instance, any finitely presented group can be realized as the fundamental group of a six dimensional symplectic CY. We also report the recent advances on symplectic log Calabi-Yau surfaces, based on joint works with Cheuk Yu Mak, Jie Min and Shengzhen Ning. These include the almost toric fibrations--log CY correspondence, a Torelli type theorem and applications to symplectic fillings, symplectic affine ruledness, symplectic Hirzebruch-Jung strings and symplectic weighted projective planes.