This talk will be about models that capture the error-prone asexual evolution of molecules such as RNA and DNA. Such models have been used to study fast-evolving organisms such as viruses and have influenced drug-design strategies which counter them. In particular, I will focus on Eigen's quasispecies model and its extensions to finite populations. I will then take a rigorous look at their underlying phenomena such as the``error-threshold" and the "speed-of-evolution", which are crucial to anti-viral approaches. Along the way I will point out some important mathematical and computational problems that arise and involve topics such as Boolean functions, threshold phenomena, and Markov chains and how the lens of theoretical computer science (theory) can be of help.