  In this work, the density in $H(b)$ spaces of finitely connected planar
domains and the boundedness of composition operators on these function spaces
are studied. Density of the algebra $\mathcal{A}(D)$ is considered for both in
the cases where the defining function $b$ is an extreme and non-extreme point
of the unit ball of $H^\infty(D)$. In the last part boundedness of composition
operators on $H(b)$ spaces is considered and as well as a generalization of the
unit disk case is given, the boundedness of composition operators with
generalized Blaschke symbols over finitely connected domains is characterized.
