  This paper proposes a construction of local $C^r$ interpolation spaces and
$C^r$ conforming finite element spaces with arbitrary $r$ in any dimension. It
is shown that if $k \ge 2^{d}r+1$ the space $\mathcal P_k$ of polynomials of
degree $\le k$ can be taken as the shape function space of $C^r$ finite element
spaces in $d$ dimensions. This is the first work on constructing such $C^r$
conforming finite elements in any dimension in a unified way. It solves a
long-standing open problem in finite element methods.
