  Sixty years ago, S. B. Myers and N. E. Steenrod ({\it Ann. of Math.} {\bf 40}
(1939), 400-416) showed that the isometry group of a Riemannian manifold
without boundary has a structure of Lie group. Recently A. V. Bagaev and N. I.
Zhukova ({\it Siberian Math. J.} {\bf 48} (2007), 579-592) proved the same
result for a Riemannian orbifold. In this paper, we firstly show that the
isometry group of a Riemannian manifold $M$ with boundary has dimension at most
${1/2} \dim M (\dim M-1)$. Then we completely classify such Riemannian
manifolds with boundary that their isometry groups attain the preceding maximal
dimension.
