We will discuss various ways to define and compute new q-series invariants that have integer powers and integer coefficients. After a quick review of the physical framework, we show how it explains and generalizes the observations of Lawrence-Zagier and Hikami et.al. to arbitrary 3-manifolds. If time permits, we will talk about a modular tensor category MTC[M3] responsible for the modularity properties of \hat Z_a (q). There are many unexpected and intriguing connections with various counting problems as well as with the works of Beliakova-Blanchet-Le and Garoufalidis-Le.