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In this paper, we study biharmonic isometric immersions of a surface into andbiharmonic Riemannian submersions from the product space $M^2\times\mathbb{R}$.Among other things, we prove that a surface with constant mean curvature $H$ in$M^2\times\mathbb{R}$ is proper biharmonic if and only if it is a part of$S^1\left(\frac{1}{2\sqrt{2}|H|}\right)\times \mathbb{R}$ in$S^2\left(\frac{1}{2|H|}\right)\times \mathbb{R}\subseteq M^2\times\mathbb{R}$.  For proper biharmonic Riemannian submersions $\pi:M^2\times \mathbb{R}\to(N^2,h)$, we prove that when the target surface is flat, then $\pi$ is locallya projection of a special twisted product, and when the target surface isnon-flat, $\pi$ is locally a special map between two warped product spaces witha warping function that solves a single ODE. As a by-product, we also provethat there is a unique proper biharmonic Riemannian submersion $H^2\times\mathbb{R}\to \mathbb{R}^2$ given by the projection of a warped product.

Biharmonic isometric immersions into and biharmonic Riemannian submersions from $M^2\times R$