1. Spacey random walks are a stochastic process that provides a probabilistic interpretation of tensor eigenvectors. The spacey random walk forgets its previous state but guesses it randomly, resulting in a limiting distribution that is a tensor eigenvector. 2. Higher-order Markov chains can be modeled as spacey random walks, which converge to tensor eigenvectors. This provides an algorithm for computing eigenvectors via numerical integration rather than algebraic methods. 3. Spacey random walks generalize Pólya urn processes and have applications in transportation modeling, clustering multi-relational data, and ranking. Learning the transition tensor from taxi trajectory data supports the spacey random walk hypothesis.