  We discuss the problem of time in quantum mechanics. In the traditional
formulation time enters the model as a~parameter, not an observable. In our
model time is a quantum observable as any other quantum quantity and it is also
a component of the spacetime position operator. In this case, instead of the
unitary time evolution, other operators, usually projection or POVM operators
which map the space of initial states into the space of final states at each
step of the evolution can be used. The quantum evolution itself is a stochastic
process. This allows to treat time as a quantum observable in a consistent,
observer independent way, which is a very important feature to resolve some
quantum paradoxes and the time problem in cosmology.
  An idea of construction of a quantum spacetime as a special set of the
allowed states is presented. An example of a structureless quantum
Minkowski-like spacetime is also considered.
  We present the projection evolution model and show how the traditional
Schroedinger evolution and relativistic equations can be obtained from it, in
the flat structureless spacetime.
  We propose the form of the time operator which satisfies the energy-time
uncertainty relation based on the same inequality as the space position and
spatial momenta observables. The sign of the temporal component of the
four-momentum operator defines the basic arrow of time in spacetime.
