  We consider the class of all sense-preserving harmonic mappings $f=
h+\overline{g}$ of the unit disk $\ID$, where $h$ and $g$ are analytic with
$g(0)=0$, and determine the Bohr radius if any one of the following conditions
holds: \bee $h$ is bounded in $\ID$. $h$ satisfies the condition ${\rm Re}\,
h(z)\leq 1$ in $\mathbb{D}$ with $h(0)>0$. both $h$ and $g$ are bounded in
$\ID$. $h$ is bounded and $g'(0)=0$. \eee We also consider the problem of
determining the Bohr radius when the supremum of the modulus of the dilatation
of $f$ in $\ID$ is strictly less than $1$. In addition, we determine the Bohr
radius for the space $\mathcal B$ of analytic Bloch functions and the space
${\mathcal B}_H$ of harmonic Bloch functions. The paper concludes with two
conjectures.
