Is there a smooth modular compactification of the moduli space of smooth curves of genus $ g > 1 $ over $ \mathbb{C} $? 


I am willing to allow for enrichments such as level structures. The compactification should be a projective variety rather than a stack. Any references are highly appreciated. 



smooth modular compactification of moduli of curves
Sure: see Eduard Looijenga, "Smooth Deligne-Mumford compactifications by means of Prym level structures", which completely answers your question.


There is also later work by de Jong-Pikaart, Boggi-Pikaart and Abramovich-Corti-Vistoli where more general non-abelian level structures are considered, and over more general base schemes.


