We provide a systematic analysis of the possible asymptotic distributions of one-dimensional continuous-time random walks (CTRWs) by applying the limit theorems of probability theory. Biased and unbiased walks of coupled and decoupled memory are considered. In contrast to previous work concerning decoupled memory and L6vy walks, we deal also with arbitrary coupled memory and with jump densities asymmetric about its mean, obtaining asymmetric L6vy-stable limits. Suprisingly, it is found that in most cases coupled memory has no essential influence on the form of the limiting distribution. We discuss interesting properties of walks with an infinite mean waiting time between successive jumps