Teaching is everywhere, yet much of what is involved in teaching remains hidden, with comprehensive theories lacking. These challenges serve as the backdrop for research on mathematics teaching and the work of Thematic Working Group 19. To make progress, the group has used four domains to organize and consider research on teaching. This paper reviews the contributions and issues that arose in the group at CERME13. We elaborate on how the domains stimulated discussions of the meaning of teaching across papers, and we provide reflections and implications for future work