  The great majority of algorithms employed in the study of lattice field
theory are based on Monte Carlo's importance sampling method, i.e. on
probability interpretation of the Boltzmann weight. Unfortunately in many
theories of interest one cannot associated a real and positive weight to every
configuration, that is because their action is explicitly complex or because
the weight is multiplied by some non positive term. In this cases one says that
the theory on the lattice is affected by the sign problem. An outstanding
example of sign problem preventing a quantum field theory to be studied, is QCD
at finite chemical potential. Whenever the sign problem is present, standard
Monte Carlo methods are problematic to apply and, in general, new approaches
are needed to explore the phase diagram of the complex theory. Here we will
review three of the main candidate methods to deal with the sign problem,
namely complex Langevin dynamics, Lefschetz thimbles and density of states
method. We will first study complex Langevin dynamics, combined with the gauge
cooling method, on the one-dimensional Polyakov line model, and then we will
apply it to pure gauge Yang-Mills theory with a topological theta-term. It
follows a comparison between complex Langevin dynamics and the Lefschetz
thimbles method on three toy models, which are the quartic model, the U(1)
one-link model with a mu dependent determinant, and the SU(2) non abelian
one-link model with complex beta parameter. Lastly, we introduce the density of
state method, based on the LLR algorithm, and we will employ it in the study of
the relativistic Bose gas at finite chemical potential.
