I have been looking for headboards recently and had not realized until now that there are a myriad of headboard designs that uses a wide variety of materials. First up, Apartment Therapy posted in January 10 Headboards one can make for under $50. Cost is definitely the constraint here so these are some very cheap or easy to obtain materials.
Burlap

Pallets

Hankies

Books

Paint

Painted Screen

Cardboard

Salvaged Wood

Ribbons
Framed Photos

And then there are some more in Apartment Therapy archives…
Corrugated Sheet Metal

Natural Wood Headboard

Vinyl Decal

Why do we need headboards? Traditionally, headboards literally hold up one end of the bed off the ground while the foot board holds up the other. As you can see above, there is a shift away from this mere function. Now not only is it decorative, some are also functional in the way that it holds decorations, or books, or general storage for duvets, blankets, sheets etc.
One thought that might be worth mentioning is that I, myself, consider a lack of bed frame somewhat strange and a quick poll with my wife and our family also reached similar results.
