In this paper the historic development of public-key cryptanalysis and important literature are surveyed.In particular the breakthroughs during 2010~2014 about the discrete logarithm problems for finite fields and elliptic curves are described.Since the birth of public key cryptography, public key cryptanalysis has been developed to be an inter discipline research with tools from mathematics, algorithmic science and experimental simulations.This survey intents to help the young researchers in Chinese cryptologic community to enter this active research field.