We study Bose-Einstein condensation (BEC) for a non-interacting gas in
a very anisotropic trap. Therefore, at enough small temperatures some
degrees of freedom ''freeze'', thus reducing the ''effective'' dimens
ionality of the system. We mainly focus in quasi-bidimensional traps c
haracterized by a surface S in the (x, y) plane. We consider two confi
ning mechanisms in the z direction: harmonic and rigid wall potentials
. There are not forces parallel to S, excepting the rigid wall at the
edges. The most relevant results are: (a) The condensate smoothly sets
at T similar to O(T-c/log N), where T-c is the tridimensional condens
ation temperature. (b) When BEC is present, also the low-lying excited
states have a macroscopic occupation this effect is also present in a
quasi-onedimensional harmonic ''cigar-shape'' trap. (c) The condensat
ion process is sensitive to the shape of S.