As of 2018,Delaware residents do not need to worry about a state estate tax, which is a tax that is levied on people who own property in the state where they died.

Delaware residents are subject to an estate tax for estates of more than $11,580,000, which is the same as the current federal estate tax exclusion. This tax falls on the estate of the person who died, the beneficiaries or heirs inherit what's left. There is no inheritance tax in Delaware, this would be tax that falls on the heirs and beneficiaries, not on the estate of the person who died.

This state-specific estate tax is in addtion to the federal estate tax, and that tax also falls on the estate of the person who died, not on the people who inherit that property. There is an exemption of $10 million, which is indexed to inflation and is currently$11,580,000, and only people who die with an estate larger than that exemption will have to pay estate tax. It's estimated that only the richest .14% of Americans will be subject to the estate tax at all, or only two out of every 1,000 people who die.

If someone dies in Delaware with less than the exemption amount (currently $11,580,000), their estate doesn't owe any federal estate tax, and if someone dies in Delaware with an estate worth less than that same amount, there is no Delaware estate tax, either. The heirs and beneficiaries inherit the property free of estate tax. They don't pay income tax on it, either, because inherited property is not ordinary income. The only exception to this are inherited retirement accounts, which are subject to income tax as the assets are withdrawn. 

Delaware estate tax return forms and instructions can be downloaded from the state’s Division of Revenue website.