The structures that attach chromosomes to spindle fibers are called kinetochores.

Here's why:

* Kinetochores are protein complexes that form on the centromere of each chromosome during cell division.

* They act as the point of attachment for spindle fibers, which are microtubules that pull chromosomes apart during mitosis and meiosis.

* Kinetochores ensure that each daughter cell receives a complete set of chromosomes.

So, the answer is kinetochores.