I am going to talk about recent topics in cryptographic theory based on elliptic and hyperelliptic curves. I will start with a general introduction on the elliptic and hyperelliptic curve cryptograpy (ECC) then explain certan recent attacks to ECC and security analysis against these attacks, including results from my own group. In particular, ECC defined over extensions of finite fields and certain new cryptosystems including isogeny based anti-quantum-computer method will be discussed.