Suppose $I$ is an index set, $(L_i)_{i \in I}$ is a family of first order languages, and $L_\cap$ is a first order language such that $L_i\cap L_j = L_\cap$ for all distinct $i,j \in I$. Suppose $T_i$ is acomplete, consistent, model complete $L_i$-theory for all $i \in I$ and suppose $T_\cap$ is an $L_\cap$-theory such that $T_i \cap T_j = T_\cap$ for all distinct $i,j \in I$. Let $T_\cup = \bigcup_{i \in I} T_i$. We discuss the question: When does $T_\cup$ have a model companion? Time permitting we will also discuss a number of motivating examples. Joint work with Minh Chieu Tran and Alex Kruckman.