Under the bankruptcy code, non exempt property is liquidated by the trustee in a Chapter 7 to provide a fund to pay claims of creditors.  In practice, however, trustees sometimes elect not to administer (that is, collect and sell) assets that have little non exempt value because the costs of doing so will consume the dollars collected.  So, the trustee may simply elect to allow the debtor to keep assets that have only a small amount of non exempt value.  The asset is deemed abandoned back to the debtor.

If a trustee elects to administer an asset that is partially exempt, the debtor may be the most obvious purchaser.  Who knows the asset better or wants it more than the debtor?  The trustee may even allow the debtor to buy out the estate's interest over a short time period.

If the trustee sells a partially exempt asset to a third party, he must pay the debtor the amount of money equal to the allowed exemption from the sale proceeds.

In Chapter 13, the debtor usually keeps all his property, exempt and non exempt.  The value of the non exempt property is used to calculate whether the proposed Chapter 13 plan meets the "best interest of creditors" test:  that is, do creditors get through the plan at least as much as they would get if the case were a Chapter 7.

-Drew Broaddus