I will discuss combinatorial aspects of some new variants of walk models, namely endless walks (which remain self-avoiding when concatenated with themselves), inverted walks (which remain self-avoiding when they are concatenated with their inversions), and timid walks (which are related to prudent walks). These models have some nice properties, and I will discuss work in progress with results from both direct solutions and computer enumerations. There are a number of open problems associated with these models which are calling out for solutions or proofs.