  Let $R$ be a ring with involution. The recently introduced notions of the
core and dual core inverse are extended from matrix to an arbitrary $*$-ring
case. It is shown that the group, Moore-Penrose, core and dual core inverse are
closely related and they can be treated in the same manner using appropriate
idempotents. The several characterizations of these inverses are given. Some
new properties are obtained and some known results are generalized. A number of
characterizations of EP elements in $R$ are obtained. It is shown that core and
dual core inverse belong to the class of inverses along an element and to the
class of $(b,c)$-inverses. The case when $R$ is algebra of all bounded linear
operators on Hilbert space is specifically considered.
