What impact does the quantum compuer has on modern cryptography?

Recently I read that IBM already has a working quantum comptuer, although small number of qubits, but that they hoped to introduce a much more powerful model within the comming years.

Also, such a capable quantum computer should be able to solve hard mathematical problems within polynomial time (if I've understood it correctly).

Is the factoring problem one of these? If so, RSA would be broken, right?

Does a quantum computer make modern crytography obsolete, so that new assumptions have to made on which to base future cryptographic schemes? (Like the factoring problem is hard-to-solve provide justification for the security of RSA).