  We investigate the free boundary Schur process, a variant of the Schur
process introduced by Okounkov and Reshetikhin, where we allow the first and
the last partitions to be arbitrary (instead of empty in the original setting).
The pfaffian Schur process, previously studied by several authors, is recovered
when just one of the boundary partitions is left free. We compute the
correlation functions of the process in all generality via the free fermion
formalism, which we extend with the thorough treatment of "free boundary
states". For the case of one free boundary, our approach yields a new proof
that the process is pfaffian. For the case of two free boundaries, we find that
the process is not pfaffian, but a closely related process is. We also study
three different applications of the Schur process with one free boundary:
fluctuations of symmetrized last passage percolation models, limit shapes and
processes for symmetric plane partitions, and for plane overpartitions.
