The authors prove some variants of the comparison principle for viscosity sub- and supersolutions of fully nonlinear elliptic equations in a domain , extending standard results. Among others, the following case is considered: at any , is strictly increasing with respect to or is non-totally degenerate, what roughly means that is a strictly decreasing function of the real parameter , with being the identity matrix and an arbitrary symmetric matrix. A further case refers to operators of the form
where is uniformly elliptic and is an matrix-valued function satisfying a non-degeneracy condition. A number of more specific equations is considered including Bellman-Isaacs equations, quasilinear subelliptic equations, and equations involving Pucci-type oerators. The results are applied to deduce existence theorems for the Dirichlet problem in the viscosity setting.