  In this work we study metric properties of the well-known family of binary
Reed-Muller codes. Let $A$ be an arbitrary subset of the Boolean cube, and
$\widehat{A}$ be the metric complement of $A$ -- the set of all vectors of the
Boolean cube at the maximal possible distance from $A$. If the metric
complement of $\widehat{A}$ coincides with $A$, then the set $A$ is called a
{\it metrically regular set}. The problem of investigating metrically regular
sets appeared when studying {\it bent functions}, which have important
applications in cryptography and coding theory and are also one of the earliest
examples of a metrically regular set. In this work we describe metric
complements and establish the metric regularity of the codes
$\mathcal{RM}(0,m)$ and $\mathcal{RM}(k,m)$ for $k \geqslant m-3$.
Additionally, the metric regularity of the codes $\mathcal{RM}(1,5)$ and
$\mathcal{RM}(2,6)$ is proved. Combined with previous results by Tokareva N.
(2012) concerning duality of affine and bent functions, this establishes the
metric regularity of most Reed-Muller codes with known covering radius. It is
conjectured that all Reed-Muller codes are metrically regular.
