The assumption that the groups follow the normal curve is the usual one made in most significance tests, though here it is somewhat stronger in that it is applied to several groups at once. Of course many distributions do not follow the normal curve, so here is one reason that ANOVA may give incorrect results. It would be wise to consider whether it is reasonable to believe that the groups' distributions follow the normal curve.
Of course the different population averages imposes no restriction on the use of ANOVA; the null hypothesis, as usual, allows us to do the computations that yield F.
The third assumption, that the populations' standard deviations are equal, is important in principle, and it can only be approximately checked by using as bootstrap estimates the sample standard deviations. In practice, statisticians feel safe in using ANOVA if the largest sample SD is not larger than twice the smallest.