we consider the construction of a polygon $p$ with $n$ vertices whose turning angles at the vertices are given by a sequence $a=(\alpha_0,\ldots, \alpha_{n-1})$, $\alpha_i\in (-\pi,\pi)$, for $i\in\{0,\ldots, n-1\}$. \\ \\ the problem of realizing $a$ by a polygon can be seen as that of constructing a straight-line drawing of a graph with prescribed angles at vertices, and hence, it is a special case of the well studied problem of constructing an \emph{angle graph}. \\ \\ in 2d, we characterize sequences $a$ for which every generic polygon $p\subset \mathbb{r}^2$ realizing $a$ has at least $c$ crossings, for every $c\in \mathbb{n}$, and describe an efficient algorithm that constructs, for a given sequence $a$, a generic polygon $p\subset \mathbb{r}^2$ that realizes $a$ with the minimum number of crossings. \\ \\ in 3d, we describe an efficient algorithm that tests whether a given sequence $a$ can be realized by a (not necessarily generic) polygon $p\subset \mathbb{r}^3$, and for every realizable sequence the algorithm finds a realization.