In this paper, we propose Gaussian random paths by defining a probability distribution over continuous paths interpolating a finite set of anchoring points using Gaussian process regression. By utilizing the generative property of Gaussian random paths, a Gaussian random path planner is developed to safely steer a robot to a goal position. The Gaussian random path planner can be used in a number of applications, including local path planning for a mobile robot and trajectory optimization for whole body motion planning. We have conducted an extensive set of simulations and experiments, showing that the proposed planner outperforms look-ahead planners which use a pre-defined subset of egocentric trajectories in terms of collision rates and trajectory lengths. Furthermore, we apply the proposed method to existing trajectory optimization methods as an initialization step and demonstrate that it can help produce more cost-efficient trajectories.