  The paper is devoted to a systematic study of the duality of processes in the
sense that $E f(X_t^x,y)=E f (x, Y_t^y)$ for a certain $f$.
  This classical topic has well known applications in interacting particles,
intertwining, superprocesses, stochastic monotonicity, exit - entrance laws,
ruin probabilities in finances, etc. Aiming mostly at the case of $f$ depending
on the difference of its arguments, we shall give a systematic study of duality
via the analysis of the generators of dual Markov processes leading to various
results and insights.
