There are many current applications of the continuous-time random walk (CTRW), particularly in describing kinetic and transport processes in different chemical and biophysical phenomena. We derive exact solutions for the Laplace transforms of the propagators for non-Markovian asymmetric one-dimensional CTRW's in an infinite space and in the presence of an absorbing boundary. The former is used to produce exact results for the Laplace transforms of the first two moments of the displacement of the random walker, the asymptotic behavior of the moments as t-->infinity, and the effective diffusion constant. We show that in the infinite space, the propagator satisfies a relation that can be interpreted as a generalized fluctuation theorem since it reduces to the conventional fluctuation theorem at large times. Based on the Laplace transform of the propagator in the presence of an absorbing boundary, we derive the Laplace transform of the survival probability of the random walker, which is then used to find the mean lifetime for terminated trajectories of the random walk.