We study completions of diagrams of extensions of C*-algebras in which all three C*-algebras in one of the rows and either the ideal or the quotient in the other are given, along with the three morphisms between them. We find universal solutions to all four of these problems under restrictions of varying severity, on the given vertical maps and describe the solutions in terms of push-outs and pull-backs of certain diagrams. Our characterization of the universal solution to one of the diagrams yields a concrete description of various amalgamated free products. This leads to new results about the K-theory of amalgamated free products, verifying the Cuntz conjecture in certain cases. We also obtain new results about extensions of matricial fieldC*-algebras, verifying partially a conjecture of Blackadar and Kirchberg. Finally, we show that almost commuting unitary matrices can be uniformly approximated by commuting unitaries when an index obstruction vanishes.