  A century ago, Einstein formulated his elegant and elaborate theory of
General Relativity, which has so far withstood a multitude of empirical tests
with remarkable success. Notwithstanding the triumphs of Einstein's theory, the
tenacious challenges of modern cosmology and of particle physics have motivated
the exploration of further generalised theories of spacetime. Even though
Einstein's interpretation of gravity in terms of the curvature of spacetime is
commonly adopted, the assignment of geometrical concepts to gravity is
ambiguous because General Relativity allows three entirely different, but
equivalent approaches of which Einstein's interpretation is only one. From a
field-theoretical perspective, however, the construction of a consistent theory
for a Lorentz-invariant massless spin-2 particle uniquely leads to General
Relativity. Keeping Lorentz invariance then implies that any modification of
General Relativity will inevitably introduce additional propagating degrees of
freedom into the gravity sector. Adopting this perspective, we will review the
recent progress in constructing consistent field theories of gravity based on
additional scalar, vector and tensor fields. Within this conceptual framework,
we will discuss theories with Galileons, with Lagrange densities as constructed
by Horndeski and beyond, extended to DHOST interactions, or containing
generalized Proca fields and extensions thereof, or several Proca fields, as
well as bigravity theories and scalar-vector-tensor theories. We will review
the motivation of their inception, different formulations, and essential
results obtained within these classes of theories together with their empirical
viability.
