Drinfeld level structures are a key concept in the arithmetic study of the moduli of elliptic curves. They also play an important role in the moduli of 1 dimensional p-divisible groups, and related Shimura varieties studied by Harris and Taylor. I'll explain why Drinfeld level structures (and the related "full set of sections" defined by Katz and Mazur) are not adequate for studying more general Shimura varieties. I'll discuss two examples of a satisfying theory of level structure outside the Drinfeld case: i) full level structures on the group $\mu_p x \mu_p$; ii) $\Gamma_1(p^r)$-type level structures on an arbitrary p-divisible group (joint work with R. Kottwitz).