Design and analysis of lightweight block ciphers have become more popular due to the fact that the future use of block ciphers in ubiquitous devices is generally assumed to be extensive. In this respect, several lightweight block ciphers are designed, of which PRESENT and HIGHT are two recently proposed ones by Bogdanov et al. and Hong et al. respectively. In this paper, we propose new attacks on PRESENT and HIGHT. Firstly, we present the first related-key cryptanalysis of 128-bit keyed PRESENT by introducing 17-round related-key rectangle attack with time complexity approximately 2^104 memory accesses. Moreover, we further analyze the resistance of HIGHT against impossible differential attacks by mounting new 26-round impossible differential and 31-round related-key impossible differential attacks where the former requires time complexity of 2^119.53 reduced round HIGHT evaluations and the latter is slightly better than exhaustive search.