  The relativistic theory of unconstrained $p$-dimensional membranes
($p$-branes) is further developed and then applied to the embedding model of
induced gravity. Space-time is considered as a 4-dimensional unconstrained
membrane evolving in an $N$-dimensional embedding space. The parameter of
evolution or the evolution time $\tau$ is a distinct concept from the
coordinate time $t = x^0$. Quantization of the theory is also discussed. A
covariant functional Schr\" odinger equations has a solution for the wave
functional such that it is sharply localized in a certain subspace $P$ of
space-time, and much less sharply localized (though still localized) outside
$P$. With the passage of evolution the region $P$ moves forward in space-time.
Such a solution we interpret as incorporating two seemingly contradictory
observations: (i) experiments clearly indicate that space-time is a continuum
in which events are existing; (ii) not the whole 4-dimensional space-time, but
only a 3-dimensional section which moves forward in time is accessible to our
immediate experience. The notorious problem of time is thus resolved in our
approach to quantum gravity. Finally we include sources into our unconstrained
embedding model. Possible sources are unconstrained worldlines which are free
from the well known problem concerning the Maxwell fields generated by charged
unconstrained point particles.
