In this work we define $\underset{\theta}{R}$-projective geodesic mappings $(\theta=1,\ldots,5)$ of two general affine connection spaces and obtain some invariant geometric objects of these mappings, generalizing Weyl's tensor. Also, we define $\underset{\theta}{R}$-projectively flat affine connection spaces $G\bar A_N$ and find necessary conditions for the space $GA_N$ to be $\underset{\theta}{R}$-projectively flat.