The main problem we will consider is whether the local lifting property (LLP) of a $C^_$-algebra implies the (global) lifting property (LP). Kirchberg showed that this holds if the Connes embedding problem has a positive solution, but it might hold even if its solution is negative. We will present several new characterizations of the lifting property for a $C^_$-algebra $A$ in terms of the maximal tensor product of A with the (full) $C^_$-algebra of the free group ${\mathbb F}_{\infty}$. We will recall our recent construction of a non-exact $C^_$-algebra with both LLP and WEP. This prompted us to try to prove that LLP implies LP for a WEP $C^*$-algebra. While our investigation is not conclusive we obtain a fairly simple condition in terms of tensor products that is equivalent to the validity of the latter implication.