Today’s parallel and distributed systems are changing in their organization and the concept of Grid computing, a set of dynamic and heterogeneous resources connected via Internet and shared by many and different users, is nowadays becoming a reality. A large number of scheduling heuristics for parallel applications described by directed acyclic graphs (DAGs) have been presented in the literature, but most of them assume a homogeneous system with a homogeneous network, i.e. a message is transmitted with the same speed on all the links. In a Grid environment this assumption cannot be done. In this thesis we tackle the problem of scheduling parallel applications described by directed acyclic graphs (DAGs) in a Grid computing system.