Recently, the old notion of causal boundary for a spacetime $V$ has been redefined consistently. The computation of this boundary $partial V$ on any standard conformally stationary spacetime $V=mathbb{R}times M$, suggests a natural compactification $M_B$ associated to any Riemannian metric on $M$ or, more generally, to any Finslerian one. The corresponding boundary $partial_BM$ is constructed in terms of Busemann-type functions. Roughly, $partial_BM$ represents the set of all the directions in $M$ including both, asymptotic and ``finite'' (or ``incomplete'') directions. This Busemann boundary $partial_BM$ is related to two classical boundaries: the Cauchy boundary $partial_{C}M$ and the Gromov boundary $partial_GM$. The authors' aims are: (1) to study the subtleties of both, the Cauchy boundary for any generalised (possibly non-symmetric) distance and the Gromov compactification for any (possibly incomplete) Finsler manifold, (2) to introduce the new Busemann compactification $M_B$, relating it with the previous two completions, and (3) to give a full description of the causal boundary $partial V$ of any standard conformally stationary spacetime.