A Sturmian sequence $x$ is defined as a cell-series of the form $\cdots B_{-1}B_0B_{1}\cdots$, where each $B_i$ is a word composed of symbols 0’s, except its first symbol which is a symbol 1, such that the difference between the symbols 0’s contained in any two $n$-chains $B_i\cdots B_{i+n-1}$ and $B_j\cdots B_{j+n-1}$ is at most one. Sturmian sequences were studied and classified by Hedlund and Morse according to their frequency: the limit $b_n / n$ as $n$ goes to infinity where $b_n$ is the number of 0’s in any given $n$-chain in $x$. In this talk we discuss the crossed products associated to the subshifts generated by Sturmian sequences. The case of crossed products of sturmian subshifts with irrational frequency coincides with the irrational rotation algebra. In the rational frequency case the Sturmian sequence is either periodic or else is a so called skew Sturmian sequence; in both cases the resulting crossed product is not simple.