One of the appealing features of tunneling as a theoretical problem is that it poses interesting questions regarding the complexified dynamics of the classical limit. Often, these questions are of a different nature to the ones that arise in a purely dynamical systems context and can therefore push the dynamics side of the problem in novel directions, without necessarily being technically hard. In this article we explore a number of scenarios where these underlying problems of complex dynamics can be well enough understood to allow explicit analytical approximations to be given for tunneling rates. The treatment here is not intended to be comprehensive and short shrift is given to very interesting and important tunneling regimes, such as those of resonance-and chaos-assisted tunneling [3,14,16,17,40,48,78], for example. Instead, by concentrating on relatively tractable problems where answers can be given in terms of simple geometrical characterizations of the complex dynamics, we aim to achieve an understanding of problems which are important in their own right but which may also later serve as building blocks in the solution of more difficult problems.