Summary: In this paper, we introduce two different generalizations of Schur numbers that involve rainbow colorings. Motivated by well-known generalizations of Ramsey numbers, we first define the rainbow Schur number \(RS(n)\) to be the minimum number of colors needed such that every coloring of \(\{1,2,\dots,n\}\), in which all available colors are used, contains a rainbow solution to \(a+b=c\). It is shown that \[ RS(n)=\lfloor\log_2(n)\rfloor+2, \text{ for all } n\geq 3. \] Second, we consider the Gallai-Schur number \(GS(n)\), defined to be the least natural number such that every \(n\)-coloring of \(\{1,2,\dots,GS(n)\}\) that lacks rainbow solutions to the equation \(a+b=c\) necessarily contains a monochromatic solution to this equation. By connecting this number with the \(n\)-color Gallai-Ramsey number for triangles, it is shown that for all \(n\geq 3\), \[ GS(n)=\begin{cases}5^k&\text{if }n=2k\\2\cdot 5^k&\text{in }n=2k+1.\end{cases} \]