Church of England 
In the established Church of England, assistant bishops are usually retired (diocesan or suffragan) bishops – in which case they are honorary assistant bishops. Occasionally active bishops are appointed to be assistant bishops – however, unlike a diocesan or suffragan they do not hold a title: they are not the "Bishop of Somewhere".
The Assistant Bishop of Leicester and the Assistant Bishop of Newcastle are assistant bishops who are active rather than retired in the Dioceses of Leicester and of Newcastle respectively. In practice, each acts almost exactly like a suffragan bishop (those Dioceses don't have one), whereas he is actually a stipendiary assistant bishop.
Anglican Church of Australia 
In the Anglican Church of Australia, the appointments of assistant bishops have been made in accordance with the Assistant Bishops' Canon since 1966. In the Australian dioceses, these assistant bishops function similarly to suffragan bishops in England (the Australian church has no suffragans per se). According to the 1966 canon, while the term coadjutor bishop may be used for an assistant bishop, no bishop may have the right or expectation to succeed to a diocesan see.
See also 
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