Diffie-Hellman in place of SSL?

Can a Diffie-Hellman key exchange algorithm be used to encrypt client-server communication on a web-page in place of SSL? If it can, what are the disadvantages (i.e. why does the standard use SSL which requires a certificate authority)? My understanding is that Diffie-Hellman can be used to secretly establish a shared key which then can be use to encrypt any further communication.