When dealing with vector fields and their smoothness we are naturally led to consider a connection (or covariant derivative) and objects such as the connection Dirichlet energy and the connection Laplacian. I will use problems from physical and geometric modeling to illustrate how a connection can encode interesting application data, often turning what look like nasty non-linear problems into optimization problems as simple as minimizing the (connection) Dirichlet energy. The subject has rich connections to many areas of physics and mathematics and I will give an introduction which does not require an advanced differential geometry background.