Patching together locally defined analytic functions to obtain a globally well defined analytic function is notoriously difficult, as the use of cutoff functions destroys analyticity. Analytic functions are solutions of thehomogeneous Cauchy--Riemann equations. Therefore, if we understand (solvability of) the inhomogeneous Cauchy--Riemann equations, we can hope to produce a correction term to restore analyticity, in such away that the desired local behavior survives. In this colloquium, I will illustrate this philosophy with several examples, including one from the classical one variable theory. In dimension more than one, one is led to consider the Cauchy--Riemann operator not just on functions, but also on forms. I will indicate the central role these considerations play in several complex variables.