Knapsack Cryptosystems are cryptosystems which security is based on the hardness of solving the knapsack problem. While such systems have been existing for quite a long time, they remain quite unpopular because a lot of such systems have been broken. However that type of cryptosystem is a good candidate for post-quantum cryptography
The most famous knapsack cryptosystem is the Merkle-Hellman Public Key Cryptosystem, one of the first public key cryptosystem, published the same year as the RSA cryptosystem. However this system has been broken by several attacks : one from Shamir, one by Adleman, and the low density attack.
However, there exist modern knapsack cryptosystems that are considered secure so far: among them is Nasako-Murakami 2006.
What is interesting with those systems is that the Knapsack problem, in the settings where no attack were found, is believed to be difficult to solve even by a quantum computer. This is not the case for systems as RSA relying on the problem of factoring big integers, a problem that is solved in linear time by Shor's quantum algorithm.