At the Oberwolfach meeting on Combinatorics on Words in 2010, the participants suggested a list of 18 important open problems in the field. In February 2018, it was updated to include some recent progress on these problems.
Here is the list:
1. Is Π Normal?
2. Frequencies in the Kolakoski word
3. Word Equation Satisfiability
4. Largest Independent System of Word Equations
7. Enumeration of Squarefree Words
8. Does there exist a k-avoidable word?
9. Is the language of primitive words context-free?
12. Reconstruction from subwords
13. The Brown-Freedman-Halbeisen-Hungerbühler-Pirillo-Varricchio Problem
15. Do Codes have unique primitive roots?
16. Is the exponent of a morphism computable?
17. Can we decide if a set of matrices generates the free group?
18. The number of right special factors
-- JeffreyShallit - 13 Oct 2010