The ability to set breakpoints stands, along with the possibility of deterministic reexecution, as one of the most important issues in the debugging of message-passing programs. We consider in this paper the design of fully distributed algorithms for the detection of breakpoints in such programs, and provide four algorithms, one for each different type of breakpoint. One of the algorithms detects the occurrence of unconditional breakpoints, while the other three detect the occurrence of breakpoints on disjunctive predicates, stable conjunctive predicates, and generic conjunctive predicates. All the algorithms we present detect breakpoints in the form of earliest global states with respect to the particular property involved. In the case of unconditional breakpoints, such an earliest global state must coincide exactly with the requested local unconditional breakpoints for the processes that do actually participate in the breakpoint. In the case of the other (conditional) breakpoints, what is detected is the earliest global state at which either the disjunctive or the conjunctive predicate under consideration is true. In order to actually halt the computation at the exact global state the algorithms detect, we suggest as a first approach the use of checkpointing and rollback-recovery techniques.