  Nudging is an important data assimilation technique where partial field
measurements are used to control the evolution of a dynamical system and/or to
reconstruct the entire phase-space configuration of the supplied flow. Here, we
apply it to the toughest problem in fluid dynamics: three dimensional
homogeneous and isotropic turbulence. By doing numerical experiments we perform
a systematic assessment of how well the technique reconstructs large- and
small-scales features of the flow with respect to the quantity and the
quality/type of data supplied to it. The types of data used are: (i) field
values on a fixed number of spatial locations (Eulerian nudging), (ii) Fourier
coefficients of the fields on a fixed range of wavenumbers (Fourier nudging),
or (iii) field values along a set of moving probes inside the flow (Lagrangian
nudging). We present state-of-the-art quantitative measurements of the
scale-by-scale {\it transition to synchronization} and a detailed discussion of
the probability distribution function of the reconstruction error, by comparing
the nudged field and the {\it truth} point-by-point. Furthermore, we show that
for more complex flow configurations, like the case of anisotropic rotating
turbulence, the presence of cyclonic and anticyclonic structures leads to
unexpectedly better performances of the algorithm. We discuss potential further
applications of nudging to a series of applied flow configurations, including
the problem of field-reconstruction in thermal Rayleigh-B\'enard convection and
in magnetohydrodynamics (MHD), and to the determination of optimal
parametrisation for small-scale turbulent modeling. Our study fixes the
standard requirements for future applications of nudging to complex turbulent
flows.
