The aim of the paper is the investigation of piecewise monotonic maps T of an interval X. The main tool is an isomorphism of (X, T) with a topological Markov chain with countable state space which is described by a 0–1-transition matrix M. The behavior of the orbits of points in X under T is very similar to the behavior of the paths of the Markov chain. Every irreducible submatrix of M gives rise to a T-invariant subset L of X such that L is the set ω(x) of all limit points of the orbit of an x∈X. The topological entropy of L is the logarithm of the spectral radius of the irreducible submatrix, which is a l^{1}-operator. Besides these sets L there are two T-invariant sets Y and P, such that for all x∈X the set ω(x) is either contained in one of the sets L or in Y or in P. The set P is a union of periodic orbits and Y is contained in a finite union of sets ω(y) with y∈X and has topological entropy zero. This isomorphism of (X, T) with a topological Markov chain is also an important tool for the investigation of T-invariant measures on X. Results in this direction, which are published elsewhere, are described at the end of the paper. Furthermore, a part of the proofs in the paper is purely topological without using the order relation of the interval X, so that some results hold for more general dynamical systems (X, T).