This paper deals with the problem of finding upper bounds on the numberof periodic solutions of a class of one-dimensional non-autonomous differentialequations: those with the right-hand sides being polynomials of degree n andwhose coeficients are real smooth 1-periodic functions. The case n = 3 givesthe so-called Abel equations which have been thoroughly studied and are quiteunderstood. We consider two natural generalizations of Abel equations. Ourresults extend previous works of Lins Neto and Panov and try to step forwardin the understanding of the case n > 3. They can be applied, as well, to controlthe number of limit cycles of some planar ordinary differential equations.