Quantum computers break all mainstream public key algorithms.
RSA, DSA, Diffie-Hellman, ElGamal, ECDSA, ECDH, X25519, Ed25519, ...
Symmetric cryptography (block/stream-ciphers, hashes) mostly not affected.
Well understood theory, but hard to engineer.
Some researchers give timeframes of 10-15 years for scalable quantum computers.
Algorithms that we believe to be resistant to quantum attacks.
Pro: Old, probably secure
Con: Large keys (~ 1 Megabyte).
Ntru, Ring-Learning-With-Errors, New Hope
Pro: Practical, fast, small keys
Con: Patents, DJB thinks security analysis is inadequate (discussion)
Pro: Very similar workflow to Diffie Hellman, small keys
Con: Not that fast, very experimental
Pro: Likely very secure
Con: Either stateful (easy to mess up) or large signatures (~ 40 Kilobytes)
Google deployed New Hope (lattice-based algorithm) key exchange in Chrome/BoringSSL and on some servers.
Hybrid key exchange with X25519: In case New Hope breaks it still has the security of X25519.
The D-Wave quantum computer can't run Shor's algorithm.
If anyone tells you that Quantum Cryptography or Quantum Key Distribution solves the problem of Quantum attacks or wants to build a Quantum Internet: This is utter nonsense. If you don't believe me please talk to me later.