  We construct the quartic version of generalized quasi-topological gravity,
which was recently constructed to cubic order in arXiv: 1703.01631. This class
of theories includes Lovelock gravity and a known form of quartic
quasi-topological gravity as special cases and possess a number of remarkable
properties: (i) In vacuum, or in the presence of suitable matter, there is a
single independent field equation which is a total derivative. (ii) At the
linearized level, the equations of motion on a maximally symmetric background
are second order, coinciding with the linearized Einstein equations up to a
redefinition of Newton's constant. Therefore, these theories propagate only the
massless, transverse graviton on a maximally symmetric background. (iii) While
the Lovelock and quasi-topological terms are trivial in four dimensions, there
exist four new generalized quasi-topological terms (the quartet) that are
nontrivial, leading to interesting higher curvature theories in $d \geq 4$
dimensions that appear well suited for holographic study. We construct four
dimensional black hole solutions to the theory and study their properties. A
study of black brane solutions in arbitrary dimensions reveals that these
solutions are modified from the `universal' properties these solutions have.
This result may lead to interesting consequences for the dual CFTs.
