I send several items a week oversees, I use a combination of people depending on what it is but for anything thats not on a pallet try either of these :

Interparcel
Mammothparcels

Depending on where its going tends to depend on what company offer the best price, I use a combination of Fed Ex, UPS but mostly DHL who tbh have never once let me down.

Everything is tracked and insured, I haven't had a item lost or broken though so couldn't comment on how to claim but I have sent some very large boxes to the US before and everything got there fine. The trick is to make sure its very well wrapped, no point putting one layer of bubble wrap round something and expect it to be protected, same goes with the boxes, pack them out with extra cardboard to stop anything punching its way out.
I probably go OTT with the wrapping but I always add a little to cover the materials and as said have never had a problem.