  In this chapter we review the recent developments of realizing $R^2$-like
inflation in the framework of a most general UV nonlocal extension of
Einstein's general theory of relativity (GR). It is a well-motivated robust
approach towards quantum gravity. In the past decades, nonlocal gravitational
theories which are quadratic in curvature have been understood to be ghost-free
and super-renormalizable around maximally symmetric spacetimes. However, in the
context of early Universe cosmology we show that one must go beyond the
quadratic curvature nonlocal gravity in order to achieve a consistent
ghost-free framework of Universe evolution from quasi de Sitter to Minkowski
spacetime. In this regard, we discuss a construction of a most general nonlocal
gravity action that leads to $R^2$-like inflation and discuss the corresponding
observational predictions for the scalar and tensor spectral tilts,
tensor-to-scalar ratio, and the primordial non-Gaussianities. We present an
analysis of how the nonlocal inflationary cosmology goes beyond the established
notions of effective field theories of inflation. Finally, we comment on some
open questions and prospects of higher curvature nonlocal gravity on its way of
achieving the UV completion.
