This is a short survey, with references, of the development in the last 40 years of important parts of random walk theory, where the leading motive is the work and influence of Frank Spitzer. A characteristic of his work was his search for random walk properties holding without a priori assumptions such as on moments.
The first important subject is fluctuation theory: ladder variables, maxima, number of positive terms, culminating in Spitzer’s formula. The combinatorial, algebraic, analytic and Wiener-Hopf methods are referred, too. The second subject is potential theory for random walks, centered around F. Spitzer’s book “Principles of random walk” (1964; Zbl 0119.343). A short sketch of the connection with classical potential theory, Markov processes, the kernel for recurrent random walk and work after “Principles of random walk” is given. Other subjects are the recurrence criteria for random walks, random walk in random environments and Spitzer’s contributions to Brownian motion, e.g. recurrence and winding number in \(R^ 2\). Two open problems are mentioned.