Despite 25 years of intensive study by mathematicians and physicists, one of the basic questions in the field remains:

How broadly do mirror manifolds exist? How do we construct mirror pairs?

Traditionally, mirror constructions relied on either on very special structure (e.g. toric mirror constructions) or additional data such as a special Lagrangian fibration or toric degeneration. I will describe how, for log Calabi-Yau manifolds with maximally degenerate boundary, one can describe the mirror partner (up to birational equivalence) intrinsically. The construction uses a classical symplectic invariant known as symplectic cohomology.

After reviewing the necessary background in symplectic topology, I will present several results showing that our mirror construction satisfies nice properties. In the toric case, I will explain how this reproduces Batyrev's mirror construction and, in more general cases, I will discuss the relation to more recent mirror constructions of Gross-Hacking-Keel and Gross-Siebert.