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We prove that any complete immersed two-sided mean convex translating soliton$\Sigma \subset \mathbb{R}^3$ for the mean curvature flow is convex. As acorollary it follows that an entire mean convex graphical translating solitonin $\mathbb{R}^3$ is the axisymmetric "bowl soliton". We also show that if themean curvature of $\Sigma$ tends to zero at infinity, then $\Sigma$ can berepresented as an entire graph and so is the "bowl soliton". Finally weclassify all locally strictly convex graphical translating solitons definedover strip regions.

Complete translating solitons to the mean curvature flow in $\mathbb{R}^3$ with nonnegative mean curvature