The possibility of considering random projections to identify probability distributions belonging to parametric families is explored. The results are based on considerations involving invariance properties of the family of distributions as well as on the random way of choosing the projections. In particular, it is shown that if a one-dimensional (suitably) randomly chosen projection is Gaussian, then the distribution is Gaussian. In order to show the applicability of the methodology some goodness-of-fit tests based on these ideas are designed. These tests are computationally feasible through the bootstrap setup, even in the functional framework. The paper includes some simulations providing power comparisons of these projections-based tests with other available tests of normality, as well as to test the Black-Scholes model for a stochastic process.