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Excision techniques are used in order to deal with black holes in numerical simulations of Einstein's equations and consist in removing a topological sphere containing the physical singularity from the numerical domain, applying instead appropriate boundary conditions at the excised surface. In this work we present recent developments of this technique in the case of constrained formulations of Einstein's equations and for spherically symmetric spacetimes. We present a new set of boundary conditions to apply to the elliptic system in the fully constrained formalism of Bonazzola et al. [Phys. Rev. D 70, 104007 (2004)], at an arbitrary coordinate sphere inside the apparent horizon. Analytical properties of this system of boundary conditions are studied and, under some assumptions, an exponential convergence toward the stationary solution is exhibited for the vacuum spacetime. This is verified in numerical examples, together with the applicability in the case of the accretion of a scalar field onto a Schwarzschild black hole. We also present the successful use of the excision technique in the collapse of a neutron star to a black hole, when excision is switched on during the simulation, after the formation of the apparent horizon. This allows the accretion of matter remaining outside the excision surface and for the stable long-term evolution of the newly formed black hole. [ABSTRACT FROM AUTHOR]