  In an ordinary three-dimensional metal the Fermi surface forms a
two-dimensional closed sheet separating the filled from the empty states.
Topological semimetals, on the other hand, can exhibit protected
one-dimensional Fermi lines or zero-dimensional Fermi points, which arise due
to an intricate interplay between symmetry and topology of the electronic
wavefunctions. Here, we study how reflection symmetry, time-reversal symmetry,
SU(2) spin-rotation symmetry, and inversion symmetry lead to the topological
protection of line nodes in three-dimensional semi-metals. We obtain the
crystalline invariants that guarantee the stability of the line nodes in the
bulk and show that a quantized Berry phase leads to the appearance of protected
surfaces states with a nearly flat dispersion. By deriving a relation between
the crystalline invariants and the Berry phase, we establish a direct
connection between the stability of the line nodes and the topological surface
states. As a representative example of a topological semimetal with line nodes,
we consider Ca$_3$P$_2$ and discuss the topological properties of its Fermi
line in terms of a low-energy effective theory and a tight-binding model,
derived from ab initio DFT calculations. Due to the bulk-boundary
correspondence, Ca$_3$P$_2$ displays nearly dispersionless surface states,
which take the shape of a drumhead. These surface states could potentially give
rise to novel topological response phenomena and provide an avenue for exotic
correlation physics at the surface.
