  A fundamental theorem of Liouville asserts that positive entire harmonic
functions in Euclidean spaces must be constant. A remarkable Liouville-type
theorem of Caffarelli-Gidas-Spruck states that positive entire solutions of
$-\Delta u=u^{ \frac {n+2}{n-2} }$, $n\ge 3$, are unique modulo M\"obius
transformations. Far-reaching extensions were established for general fully
nonlinear conformally invariant equations through the works of
Chang-Gursky-Yang, Li-Li, Li, and Viaclovsky. In this paper, we derive
necessary and sufficient conditions for the validity of such Liouville-type
theorems. This leads to necessary and sufficient conditions for local gradient
estimates of solutions to hold, assuming a one-sided bound on the solutions,
for a wide class of fully nonlinear elliptic equations involving Schouten
tensors. A pivotal advancement in proving these Liouville-type theorems is our
enhanced understanding of solutions to such equations near isolated
singularities. In particular, we utilize earlier results of
Caffarelli-Li-Nirenberg on lower- and upper-conical singularities. For general
conformally invariant fully nonlinear elliptic equations, we prove that a
viscosity super- (sub-)solution can be extended across an isolated singularity
if and only if it is a lower- (upper-)conical singularity. We also provide
necessary and sufficient conditions for lower- (upper-)conical behavior of a
function near isolated singularities in terms of its conformal Hessian.
