  We consider different problems within the general theme of long-range
percolation on oriented graphs. Our aim is to settle the so-called truncation
question, described as follows. We are given probabilities that certain
long-range oriented bonds are open; assuming that the sum of these
probabilities is infinite, we ask if the probability of percolation is positive
when we truncate the graph, disallowing bonds of range above a possibly large
but finite threshold. We give some conditions in which the answer is
affirmative. We also translate some of our results on oriented percolation to
the context of a long-range contact process.
