Both Cambridge City Council and South Cambridgeshire District Council (SCDC) (see councils for more on how these fit together) run a household waste collection scheme using green and black bins with the bins being emptied on alternate weeks.
In addition to this they operate a box scheme using plastic boxes (black in the City, green in SCDC) which are used to collect re-cyclable refuse.
Exactly what they will collect via this method has varied over time and is quite subtle at the edges (for example whether or not window envelopes are accepted) so rather than trying to track what the councils will accept on this page you should really visit the relevant page on the councils' own web sites.
The City scheme is detailed here.
(These pages have a nasty habit of changing URL so if you find the links above broken then please take a second to find the new location and update this page. Thank you.)