Summary: In various frameworks, to assess the joint distribution of a \(k\)-dimensional random vector \(X = (X_1, \dots, X_k)\), one selects some putative conditional distributions \(Q1, \dots, Qk\). Each \(Q_i\) is regarded as a possible (or putative) conditional distribution for \(X_i\) given \((X_1, \dots, X_{i-1},\newline X_{i+1}, \dots, X_k)\). The \(Q_i\) are compatible if there is a joint distribution \(P\) for \(X\) with conditionals \(Q_1, \dots, Q_k\). Three types of compatibility results are given in this paper. First, the \(X_i\) are assumed to take values in compact subsets of \(\mathbb R\). Second, the \(Q_i\) are supposed to have densities with respect to reference measures. Third, a stronger form of compatibility is investigated. The law \(P\) with conditionals \(Q_1, \dots, Q_k\) is requested to belong to some given class \(\mathcal P_0\) of distributions. Two choices for \(\mathcal P_0\) are considered, that is, \(\mathcal P_0 = \text{exchangeable laws}\) and \(\mathcal P_0 = \text{laws with identical univariate marginals}\).