Bespopovtsy is one of the two major strains of Old Believers, the one that rejects priests and a number of church rites, such as the Eucharist

Old Believers
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The reasoning behind this is a belief that any priest or hierarch who has used the Nikonian Rites has forfeited Apostolic Succession.

Patriarch Nikon Revising Service-Books
pinterest button Patriarch Nikon Revising Service-Books Aleksey Danilovich Kivshenko (1851–1895), Public Domain

There are several confessions of them: Pomortsy, Fedoseyans, Filippians, Beguny («Runners»), Netovtsy/Netovshchina, and many others.

Old Believers
pinterest button Old Believers   ZwieRys, GNU 1.2

Bespopovtsy are Old Believers that reject Nikonian priests, however, they still believe in the existence of a Priesthood; but happen to find themselves without priests.

Historically the Bespopovtsy have received Nikonian priests that have publicly repented from the reforms of Patriarch Nikon.

Useful Information

Russian: Безпоповцы
Lit. ‘priestless'