  We study orthogonal structures and Fourier orthogonal series on the surface
of a paraboloid $\mathbb{V}_0^{d+1} = \{(x,t): \|x\| = \sqrt{t}, \, x \in
\mathbb{R}^d, \, 0\le t<1\}$. The reproducing kernels of the orthogonal
polynomials with respect to $t^\beta(1-t)^\gamma$ on $\mathbb{V}_0^{d+1}$ are
related to the reproducing kernels of the Jacobi polynomials on the parabolic
domain $\{(x_1,x_2): x_1^2 \le x_2 \le 1\}$ in $\mathbb{R}^2$. This connection
serves as an essential tool for our study of the Fourier orthogonal series on
the surface of the paraboloid, which allow us, in particular, to study the
convergence of the Ces\`aro means on the surface. Analogous results are also
established for the solid paraboloid bounded by $\mathbb{V}_0^{d+1}$ and the
hyperplane $t=1$.
