  We give a survey on the development of the study of the asymptotic Dirichlet
problem for the minimal surface equation on Cartan-Hadamard manifolds. Part of
this survey is based on the introductory part of the doctoral dissertation of
the author. The paper is organised as follows. First we introduce
Cartan-Hadamard manifolds and the concept of asymptotic Dirichlet problem, then
discuss about the development of the results and describe the methods used in
the proofs. In the end we mention some results about the nonsolvability of the
asymptotic Dirichlet problem.
