A differentiable function $f:{\bbfR}\sp n\to {\bbfR}$ is said to be invex if there exists a function $\eta (y,x)\in {\bbfR}\sp n$ such that, for all $y,x\in {\bbfR}\sp n$ $$ f(y)-f(x)\ge \eta (y,x)\sp t\nabla f(x). $$ Various extensions of such functions including pseudo- and quasi-invex have been defined and their relationship to each other and other generalizations of convexity have been studied. For the non- differentiable case, f is said to be pre-invex if $$ f(x+t\eta (y,x))\le tf(y)+(1-t)f(x),\quad 0\le t\le 1. $$ This comprehensive paper brings together many of these scattered results which are then studied, compared, and extended. Some definitions that are introduced include $\eta$-invex subsets, pseudo and quasi-pre-invex functions. One very minor correction. Reference 2 should be to the Journal, not the Bulletin, of the Australian Mathematical Society.