A matrix $A^-$ is a generalized inverse of matrix $A$ if $AA^-A=A$ holds. Let $G$ be a function which assigns to each matrix $A$ the specific subset of the set of all generalized inverses of $A$. Then we say that $A$ is below matrix $B$ if $AA^-=BA^-$ and $A^-A=A^-B$ for some $A^-\in G(A)$. The minus, star, sharp, core and dual core partial orders are some of the well known matrix partial orders defined by appropriate choices of function $G$. This article reviews the recent known results concerning generalizations of matrix partial orders to the setting of arbitrary rings with or without involution. The article mainly consists of the published results of the authors.