The Vernam cipher inspired two directions of cryptographic innovation:
Pseudo-Random Keys (stream ciphers)
Variable, Unrestricted Size Keys (Daniel)
Stream ciphers construct a random-looking key of sufficiently large size, but one that is actually computed from a very short seed, which is the real key.
Ciphers with unrestricted variable size keys are such that their computational burden is not unduly increased with the size of the key. When the size of the key approaches the size of the plaintext equivocation comes into play.