I will discuss the problem of a quantum many-body system evolving under its own Hamiltonian and subject to local weak quantum measurements. The resulting quantum many-body trajectories evolve according to a stochastic Schrodinger equation. Averaging the wave function over the realisations of the stochastic process gives rise to a Lindblad master equation for the density matrix, which often leads to heating and to featureless stationary states. However, hidden in the random fluctuations of the measuring process there is a much richer physics - with new dynamical phases characterised by qualitatively different entanglement properties and sharp phase transitions between them -which has started to be explored only very recently. I will discuss this measurement-induced criticality in the context of a Quantum Ising Chain, which I will show to display a transition from a critical phase with logarithmic scaling of the entanglement entropy to an area-law phase corresponding to the onset of the Quantum Zeno Effect. I will argue how the essential features of this problem can be understood by looking at the associated non-Hermitian Hamiltonian and its spectral transition. I will substantiate this idea with a phenomenological quasiparticle picture for the entanglement growth in systems with measurements.