  A strict monoidal category referred to as affine Brauer category
$\mathcal{AB}$ is introduced over a commutative ring $\kappa$ containing
multiplicative identity $1$ and invertible element $2$. We prove that morphism
spaces in $\mathcal{AB}$ are free over $\kappa$. The cyclotomic (or level $k$)
Brauer category $\mathcal{CB}^f(\omega)$ is a quotient category of
$\mathcal{AB}$. We prove that any morphism space in $\mathcal{CB}^f(\omega)$ is
free over $\kappa$ with maximal rank if and only if the $\mathbf u$-admissible
condition holds in the sense of (1.30). Affine Nazarov-Wenzl algebras and
cyclotomic Nazarov-Wenzl algebras will be realized as certain endomorphism
algebras in $\mathcal{AB}$ and $\mathcal{CB}^f(\omega)$, respectively. We will
establish higher Schur-Weyl duality between cyclotomic Nazarov-Wenzl algebras
and parabolic BGG categories $\mathcal O$ associated to symplectic and
orthogonal Lie algebras over the complex field $\mathbb C$. This enables us to
use standard arguments in [1,26,27] to compute decomposition matrices of
cyclotomic Nazarov-Wenzl algebras. The level two case was considered by Ehrig
and Stroppel in [14].
