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It is not true, as some Internet sites say, that Delaware has abolished its counties. It is true that property taxation was originally based on the hundred, a Saxon unit in early England, instead of counties. Today, though, property taxes are levied by the municipality (if the property is in a city or town) and the county and the school district the property is in.
The hundred still legally exists; in fact, there are 33 hundreds in the state, the boundaries of all of which are fixed by law. However, the hundred is largely an anachronism now, the primary purpose of which seems to be to show up on property deeds to identify the location of the parcel being transferred.
Each county also has its own elected executive officials and legislative body. However, many functions which in larger states are handled by individual counties -- such as law-enforcement, the judiciary, road maintenance, etc. -- have in Delaware been assumed by the state government.
Law-enforcement services in unincorporated areas of the state (i.e., most of the state) are handled by the Delaware State Police (and, in New Castle County, the New Castle County Police). The sheriffs of the three counties have no official law-enforcement duties other than service of legal process and holding prisoners at the courthouse. Even courthouse security duties have been assumed by the Sussex County, is currently embroiled in a legal battle with the Sussex County Council and the Delaware Attorney General over his authority to equip his deputies' vehicles with emergency lights and sirens. He claims that the Pennsylvania, where it took a decision of the Pennsylvania Supreme Court and an act of the General Assembly of Pennsylvania to give such powers to sheriff's deputies and constables.)