Indexable parting and grooving tools are used with lathes or CNC turning machines to remove sections of a workpiece in parting or cut-off operations or add grooves to a workpiece. They have a tool body or toolholder that accepts indexable inserts or modular heads to form a cutting edge. They require fewer tool changes than solid tools in high-volume metalworking and fabrication applications with high speeds, high feeds, and difficult-to-machine materials. The inserts and heads can be swapped out for new ones without removing the tool body or toolholder from the machine. Indexable parting and cut-off blades have a slim profile to make deep cuts. Indexable parting and grooving toolholders hold an indexable insert or cut-off blade. Indexable parting and grooving inserts can be rotated (indexed) to expose a fresh cutting edge when the old one dulls. Indexable cutting unit heads are used as part of modular quick-change tool system.