A time-dependent collision probability method has been developed for the solution of neutron transport and nuclear reactor kinetics problems in one-dimensional slab geometry. The time-dependent collision probabilities permit the solution of time-dependent neutron transport problems involving general source distributions over an indefinite time period and an infinite number of collision generations. The method is based on the analytic integration of the time-dependent integral transport kernel involving purely real cross sections. The neutron time-of-flight and causality considerations lead to a number of complex formulas involving exponential and exponential integral functions. Occasional conflicts between the regular grid in time and space and the causality considerations lead to some formulas that are inexact. It is shown that these inexact formulas are terms of the third order in the time-step length, and thus the method has overall second-order accuracy in time. The method has been used to solve two types of neutron transport problems. The first, a pulsed, planar, fixed-source problem, yielded a flux solution with a root-mean-square relative difference of 0.94% from a benchmark analytic solution. The second problem solved was a pair of multigroup nuclear reactor kinetics problems. While the kinetics results were not conclusive, they suggest that diffusion theory may yield results that underestimate the amplitude and deposited energy of certain reactor transients.