What municipalities do I have to file local returns for in Ohio? I have earned income in Akron and Toledo and currently live in Maumee.

[Hal_Al]

Simple answer: Maumee, only.

 You normally only have to file a return in the city you live in. It is not necessary to file one in the city you work in, if different. Your employer will remit withholding to the city you work in, but you do not file a return unless you're claiming a refund or your employer didn't withhold enough ( both highly unusual). The city you live in will allow a credit or partial credit for the withheld tax you paid to the work location city. 

[Hal_Al]

If you moved during 2018, you have to file a city return for each city you lived in.  If you were self employed or your employer failed to withhold for a work place city, you have to file a return for the work city, but only if you worked in that city more than 20 days.