Is it decidable to determine if a given shape can tile the plane?
According to the introduction of [1],


The complexity of determining if a single polyomino tiles the plane remains open [2,3], and
There is an undecidability proof for sets of 5 polyominoes [4].


[1] Stefan Langerman, Andrew Winslow. A Quasilinear-Time Algorithm for Tiling the Plane Isohedrally with a Polyomino. ArXiv e-prints, 2015. arXiv:1507.02762 [cs.CG]

[2] C. Goodman-Strauss. Open questions in tiling. Online, published 2000.

[3] C. Goodman-Strauss. Can’t decide? undecide! Notices of the American
Mathematical Society, 57(3):343–356, 2010.

[4] N. Ollinger. Tiling the plane with a fixed number of polyominoes. In
A. H. Dediu, A. M. Ionescu, and C. Mart´ın-Vide, editors, LATA 2009,
volume 5457 of LNCS, pages 638–647. Springer, 2009.