Organ donation after death, are we as Muslims permitted to become organ donators and if not why?

I recently watched a bbc documentary which followed a young woman who suffered from cystic fibrosis in the days around her wedding day and then organ transplant which saved her life. It just got me wondering whether Muslims are allowed to become organ donators after death and whether of not some Muslims say yes and others say no. I would also like to hear about blood donations. I hope that someone can answer these questions of mine.