Oblivious Decryption: Decrypting with a private key, without knowing the message
This can be done by exploiting the homomorphic property of RSA.  Let's say Alice's key is $(e,N,d)$ where $e$ is the public exponent, $N$ the modulus, and $d$ the private exponent.

To decrypt $x$, Bob samples $r$ randomly from $\{1,\cdots,N\}$ and computes $xr^e\mod N$ and sends it to Alice.  Alice computes $(xr^e)^d=x^dr\mod N$ and sends it back to Bob.  Bob multiplies by $r^{-1}$ to extract $x^d$.

Note that Alice does not know $r$ and thus cannot determine $x^d$ from $x^dr$.  This does lead to at least one vulnerability: Bob can send Alice whatever he likes.  In particular, he could ask for the decryptions of other private messages to Alice that he somehow acquired.