Distributed programs are particularly vulnerable to software faults. Bugs in these programs are usually very hard to detect without automatic verification. The idea of checking an expected property in a given distributed program run (also referred to as runtime verification) has recently been attracting a great deal of attention for analyzing execution traces to ensure the reliability and dependability of distributed programs.  Due to concurrency, the number of global states of a distributed program run tends to grow exponentially with respect to the number of program statements executed. As a result, checking the satisfaction of a property in a given distributed program run can incur significant overhead. This thesis introduces various ideas and exploits them to develop efficient dynamic property checking algorithms. These include the use of atom, introducing and exploiting the notion of serialization and finally proposing a methodology that exploits the concept of atoms and partial order semantics to specify and to check properties of distributed programs.  The abstract specification of a distributed program can be mapped to the lower level implementation by labeling the code blocks that belong to the abstract functionalities of the program that are expected to be performed atomically. Each labeled code block is called an atom. Dynamically, an atom includes all the events that result from executing the selected statements from the corresponding code block. An efficient on-the-fly atomicity error detection algorithm has been developed. It is shown that if a run of a distributed program is atomic then the required properties can be checked on a reduced lattice, referred to as the atomic state lattice, which is significantly smaller than the original state lattice.  Even with atomization, the number of global states can still grow exponentially in the number of atoms executed. However, when a number of processes has to maintain a property, we expect that each process will be, at some point in time, aware of the events of other processes that may affect the property. Consequently, it is not necessary to check the property in each state. Only synchronized states, where processes have already exchanged the information necessary to maintain the property, need to be considered. These states can be characterized by a synchronization predicate. Serialization of synchronized states is the minimal avenue for a set of processes to exchange the necessary information to maintain a property. Two efficient algorithms to check the satisfaction of a property in a distributed computation in cases where the synchronization predicate is conjunctive or disjunctive have been developed.  Finally, a methodology based on the concept of atoms and a partially ordered multi-set (POMSET) model to specify and to check distributed programs properties has been proposed. The POMSET model promotes the separation of two different concerns in specifying and checking properties, namely, the ordering requirements and the computational requirements. A methodology to specify and to efficiently check the two requirements has been introduced.