A tangent process of a random process X at a point z is defined to be the limit in distribution of some sequence of scaled enlargements of X about z. The main result of the paper is that a tangent process must be self-similar with stationary increments, at almost all points z where the tangent process is essentially unique. The consequences for tangent processes of certain classes of process are examined, including stable processes and processes with independent increments where unique tangent processes are Lévy processes.