Theoretical monochromatic light curves and photospheric expansion velocities are compared with observations of 27 Type Ia supernovae (SNe Ia). A set of 37 models has been considered which encompasses all currently discussed explosion scenarios for Type Ia supernovae including deflagrations, detonations, delayed detonations, pulsating delayed detonations and tamped detonations of Chandrasekhar mass, and Helium detonations of low mass white dwarfs. The explosions are calculated using one-dimensional Lagrangian hydro and radiation-hydro codes with incorporated nuclear networks. Subsequently, light curves are constructed using our LC scheme which includes an implicit radiation transport, expansion opacities, a Monte-Carlo γ-ray transport, and molecular and dust formation. For some supernovae, results of detailed non-LTE calculations have been considered.Observational properties of our series of models are discussed, in particular, the relation between the absolute brightness, post-maximum decline rates, the colors at several moments of time, etc. All models with a 56 Ni production larger than ≈ 0.4M ⊙ produce light curves of similar brightness. The influence of the cosmological red shift on the light curves and on the correction for interstellar reddening is discussed.Based on data rectification of the standard deviation, a quantitative procedure to fit the observations has been used to the determine the free parameters, i.e. the distance, the reddening, and the time of the explosion. Fast rising light curves (e.g., SN 1981B and SN 1994D) can be reproduced by delayed detonation models or deflagration models similar to W7. Slowly rising (t max ≥ 16 days) light curves (e.g., SN 1984A and SN 1990N) cannot be reproduced by standard detonation, deflagration, or delayed detonation models. To obtain an acceptable agreement with observations, models are required where the C/O white dwarf is surrounded by an unburnt extended envelope of typically 0.2 to 0.4 M ⊙ which may either be pre-existing or produced during the explosion. Our interpretation of the light curves is also supported by the photospheric expansion velocities. Mainly due to the fast increase of the γ radiation produced by the outer 56 Ni , the post maximum decline of Helium detonations tends to be faster compared to observations of normal bright SNe Ia.Strongly subluminous SNe Ia can be understood in the framework of pulsating delayed detonations, both from the absolute brightness and the colors. Alternatively, subluminosity can be produced within the scenario of helium detonations in low mass white dwarfs of about 0.6 to 0.8 M ⊙ if the explosion occurs when rather little Helium has been accreted. However, even subluminous Helium detonation models are very blue at maximum light owing to heating in the outer layers and brighter models show a fast postmaximum decline, in contradiction to the observations. 3We find evidence for a correlation between the type of host galaxy and the explosion mechanism. In spiral galaxies, about the same amount of prompt exp...