A fundamental problem in computer simulation of systems of biophysical interest is the separation of timescales. This refers to the fact that stable integration of the equations that describe molecular motion requires timesteps on the order of the fastest motion, on the order of a few femtoseconds, while events of interest, such as ligand binding, protein conformational rearrangements, etc. take place on timescales of milliseconds or more. Since even the fastest computers can barely reach the millisecond timescale by brute force, a naive computation of most `textbook' quantities of interest such as on and off rates for ligands or free energy differences fail due to sampling error. The main focus of this dissertation is to develop novel computational methods to attack the rare-event problem and compute such quantities and more. Our approach throughout is general: for each of the methods, we attempt to make as few assumptions on the underlying dynamics beyond Markovianity, while at the same time attempting to compute as broad a class of statistics as possible. To this end, our main tool is unbiased short trajectories. We generate a swarm of unbiased, short trajectories, each of which may be run in parallel, and then develop algorithms to recombine these short trajectories to compute our kinetic statistics of interest. We begin with a simple linear basis expansion method known as the dynamical Galerkin approximation. We reformulate the method to account for finite lag times and also develop new estimators for the reaction rate and reactive current. Next, we introduce two machine learning based alternatives to the DGA method. These use the universal function approximation properties of certain neural networks to overcome limitations in the basis set construction required for DGA. Finally, we combine our short trajectory methods with the weighted ensemble path sampling scheme to develop a rapidly converging algorithm for computing arbitrary stationary distributions for Markov processes. We illustrate our methods on well-characterized low-dimensional test systems, as well as models of both subseasonal weather and protein folding.