  This paper proposes a construction of $C^r$ 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.
