The paper analyzes, in the context of two person games, alternative notions of approximate common knowledge. One individual `\(p\)-believes' the event \(E\) if he assigns it a probability at least \(p\). \(E\) is `common \(p\)-belief' if both individuals \(p\)-believe \(E\), both \(p\)-believe that both \(p\)-believe \(E\) etc. \(E\) is `iterated \(p\)-belief' for individual 1 if 1 \(p\)-believes \(E\),1 \(p\)-believes that 2 \(p\)-believes \(E\), 1 \(p\)-believes that 2 \(p\)-believes that 1 \(p\)-believes \(E\) etc. \(E\) is iterated \(p\)-belief if \(E\) is iterated \(p\)-belief for both individuals. Another definition of approximate common knowledge explored is that of `weak common \(p\)-belief'. \(E\) is weak common \(p\)-belief if it is common \(p\)-belief either given individuals' actual information or if individuals ignore some of their information. The paper shows that the nature of approximate common knowledge is sensitive to the exact definition of common knowledge in such a framework where `knowledge' is replaced by lief with high probability. When \(p =1\), all three concepts are equivalent to each other as well as to the standard definitions of common knowledge, however, it is shown that when \(p< 1\) they are not equivalent but common \(p\)-belief implies iterated \(p\)-belief and weak common \(p\)-belief. Moreover, it is shown that the conjecture that for any \(p< 1\), there should exist some \(q\) sufficiently close to 1 such that if \(E\) is iterated \(q\)-belief then it must be comeon \(p\)-belief' is false. The paper also provides results relating each of the alternative definitions of approximate common knowledge to a class of applications. It is known that common \(p\)-belief is the natural notion of approximate common knowledge when studying the robustness of equilibria to a lack of common knowledge of payoffs. It is shown in this paper that iterated \(p\)-belief is the relevant notion for the study of best response dynamics in incomplete information games while weak common \(p\)-belief is the relevant notion for aggreeing to disagree/no trade type results. A striking feature of common \(p\)-belief is that improved information may reduce the degree of common \(p\)-belief. More information is not necessarily better for achieving coordination in strategic environments because common \(p\)-belief is what matters in achieving coordination in strategic environments.