As embodied in the title of the paper strong and weak variants of continuity that lie strictly between strong continuity of Levine and almost continuity due to Singal and Singal are considered. Basic properties of almost completely continuous functions (≡ R-maps) and δ-continuous functions are studied. Direct and inverse transfer of topological properties under almost completely continuous functions and δ-continuous functions are investigated and their place in the hier- archy of variants of continuity that already exist in the literature is out- lined. The class of almost completely continuous functions lies strictly between the class of completely continuous functions studied by Arya and Gupta (Kyungpook Math. J. 14 (1974), 131-143) and δ-continuous functions defined by Noiri (J. Korean Math. Soc. 16, (1980), 161-166). The class of almost completely continuous functions properly contains each of the classes of (1) completely continuous functions, and (2) al- most perfectly continuous (≡ regular set connected) functions defined by Dontchev, Ganster and Reilly (Indian J. Math. 41 (1999), 139-146) and further studied by Singh (Quaestiones Mathematicae 33(2)(2010), 1–11) which in turn include all δ-perfectly continuous functions initi- ated by Kohli and Singh (Demonstratio Math. 42(1), (2009), 221-231) and so include all perfectly continuous functions introduced by Noiri (Indian J. Pure Appl. Math. 15(3) (1984), 241-250).