To add to Keith's answer, there are various classes of number fields which are
known to be not monogenic. For instance, the following paper 

Marie-Nicole Gras,
Non monogénéité de l'anneau des entiers des extensions cycliques de $\mathbb{Q}$
de degré premier $l\ge 5$, *J. Number Theory*  **23** (1986), 347-353

gives an elegant proof of the fact that no cyclic extension $K$ of
the rationals of prime degree $l\ge 5$ is monogenic unless it happens
to be the real part of a cyclotomic field.