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When a quantum system evolves under unitary dynamics, as produced by either a Hamiltonian or by a sequence of gates inside a quantum computer, its various component parts tend to become more entangled with each other. Making measurements, on the other hand, tends to reduce this entanglement by collapsing some of the system's degrees of freedom. In this talk I'll consider what happens to the entanglement when a quantum many-body system undergoes both unitary evolution and sporadic measurements. I'll show that the competition between these two effects leads to a new kind of dynamical phase transition, such that when the measurement rate is lower than a critical value the dynamics is "entangling", while a higher-than-critical measurement rate leads to a "disentangling" phase. I will discuss our work demonstrating the existence of this transition, as well as more recent efforts to find exact solutions for its critical properties. I will show how intuition from classical percolation in disordered conductors played a key role in our understanding of the transition.