We study the character variety of representations of the fundamental group of a closed surface of genus ${g\geq 2}$ into the Lie group $SO(n,n+1)$ using Higgs bundles. For each integer $0<d\leq n(2g-2),$ we show there is a smooth connected component of the character variety which is diffeomorphic to the product of a certain vector bundle over a symmetric product of a Riemann surface with the vector space of holomorphic differentials of degree $2,4,....,2n-2.$ In particular, when $d=n(2g-2)$, this recovers Hitchin's parameterization of the Hitchin component. We also exhibit $2^{2g+1}-1$ additional connected components of the $SO(n,n+1)$-character variety and compute their topology.
Moreover, representations in all of these new components cannot be continuously deformed to representations with compact Zariski closure. Using recent work of Guichard-Wienhard on positivity, it is shown that each of the representations which define singularities (i.e., those which are not irreducible) in these $2^{2g+1}-1$ connected components are positive Anosov representations.