We will discuss our recent work on the nature of quantum dynamics and non-equilibrium phase transitions in random quantum circuits. The first half of the talk will focus on the early stages of the approach to thermal equilibrium in isolated quantum systems. By utilizing the statistical properties of the entanglement spectra, we reveal three distinct time scales that characterize the approach to thermal equilibrium. While our main focus is on random quantum circuits, we will also show how these features manifest in chaotic Hamiltonian dynamics, thus being a generic property of the approach to equilibrium in many-body quantum systems. In the second half of the talk we will discuss the recently discovered measurement driven quantum phase transition. We consider a Haar random circuit and make local projective measurements at each site with a probability p , which drives the time-dependent wave-function through a phase transition from a volume law entangled phase to area law entangled. Previous work, while being able to identify the existence of transition, had trouble capturing the critical properties accurately due to a poor choice of scaling variable. We circumvent these issues by introducing the tri-partite mutual information and an order parameter, which both provide an accurate estimate of the critical measurement rate p c and the corresponding critical exponents. Lastly, the universality class of the transition that is constrained by our numerical results will be discussed.