If
(
A
,
ρ
,
G
)
(A,\rho ,G)
is a covariant system over a locally compact group G, i.e.
ρ
\rho
is a homomorphism from G into the group of
∗
^{\ast }
-automorphisms of an operator algebra A, there is a new operator algebra
A
\mathfrak {A}
called the covariance algebra associated with
(
A
,
ρ
,
G
)
(A,\rho ,G)
. If A is a von Neumann algebra and
ρ
\rho
is
σ
\sigma
-weakly continuous,
A
\mathfrak {A}
is defined such that it is a von Neumann algebra. If A is a
C
∗
{C^{\ast }}
-algebra and
ρ
\rho
is norm-continuous
A
\mathfrak {A}
will be a
C
∗
{C^{\ast }}
-algebra. The following problems are studied in these two different settings: 1. If
A
\mathfrak {A}
is a covariance algebra, how do we recover A and
ρ
\rho
? 2. When is an operator algebra
A
\mathfrak {A}
the covariance algebra for some covariant system over a given locally compact group G?