They usuallly start on the first of a given month. However, sometimes a challange starts late, and since 25 days is alotted for challenges it might go a bit into the next month. Since most people only require a week to create a map (many in much less time than that), you can start now and participate.
There is no hard start date of a challenge, it varies. But around the 1st of the month the normal challenges start. For Lite challenges it's around the 15th.