A regular cardinal kappa has the tree property if and only if every kappa-tree (tree of height kappa with every level of size less than kappa) has a cofinal branch. This is a basic compactness/reflection property in combinatorial set theory.

A long standing open problem asked whether it is consistent that there is a singular lambda such that lambda-plus has the tree property and the Singular Cardinals Hypothesis fails at lambda: this was recently settled (positively) by Neeman. I'll discuss the background, sketch Neeman's solution, and discuss some open problems.