  We define dynamical universality classes for many-body systems whose unitary
evolution is punctuated by projective measurements. In cases where such
measurements occur randomly at a finite rate $p$ for each degree of freedom, we
show that the system has two dynamical phases: `entangling' and
`disentangling'. The former occurs for $p$ smaller than a critical rate $p_c$,
and is characterized by volume-law entanglement in the steady-state and
`ballistic' entanglement growth after a quench. By contrast, for $p > p_c$ the
system can sustain only area-law entanglement. At $p = p_c$ the steady state is
scale-invariant and, in 1+1D, the entanglement grows logarithmically after a
quench.
  To obtain a simple heuristic picture for the entangling-disentangling
transition, we first construct a toy model that describes the zeroth R\'{e}nyi
entropy in discrete time. We solve this model exactly by mapping it to an
optimization problem in classical percolation.
  The generic entangling-disentangling transition can be diagnosed using the
von Neumann entropy and higher R\'{e}nyi entropies, and it shares many
qualitative features with the toy problem. We study the generic transition
numerically in quantum spin chains, and show that the phenomenology of the two
phases is similar to that of the toy model, but with distinct `quantum'
critical exponents, which we calculate numerically in $1+1$D.
  We examine two different cases for the unitary dynamics: Floquet dynamics for
a nonintegrable Ising model, and random circuit dynamics. We obtain compatible
universal properties in each case, indicating that the entangling-disentangling
phase transition is generic for projectively measured many-body systems. We
discuss the significance of this transition for numerical calculations of
quantum observables in many-body systems.
