The theory of phase transitions represents a central concept for the characterization of equilibrium matter. In this work we study experimentally an extension of this theory to the nonequilibrium dynamical regime termed dynamical quantum phase transitions (DQPTs). We investigate and measure DQPTs in a string of ions simulating interacting transverse-field Ising models. During the nonequilibrium dynamics induced by a quantum quench we show for strings of up to 10 ions the direct detection of DQPTs by revealing nonanalytic behavior in time. Moreover, we provide a link between DQPTs and the dynamics of other quantities such as the magnetization, and we establish a connection between DQPTs and entanglement production. DOI: 10.1103/PhysRevLett.119.080501 Today, the equilibrium properties of quantum matter are theoretically described with great success. Yet, in recent years pioneering experiments have created quantum states beyond this equilibrium paradigm [1,2]. Thanks to this progress, it is now possible to experimentally study exotic phenomena such as many-body localization [3,4], prethermalization [5,6], particle-antiparticle production in the lattice Schwinger model [7], and light-induced superconductivity [8]. Understanding general properties of such nonequilibrium quantum states provides a significant challenge, calling for concepts that extend important principles such as universality to the nonequilibrium realm. A general approach towards this major goal is the theory of dynamical quantum phase transitions (DQPTs) [9], which extends the concept of phase transitions and thus universality to the nonequilibrium regime. In this Letter, we directly observe the defining real-time nonanalyticities at DQPTs in a trapped-ion quantum simulator for interacting transverse-field Ising models. Moreover, we provide a link between DQPTs and the dynamics of other relevant quantities such as the magnetization, and we establish a connection between DQPTs and entanglement production.Statistical mechanics and thermodynamics provide us with an excellent understanding of equilibrium quantum many-body systems. A key concept in this framework is the canonical partition function ZðTÞ ¼ Trðe −H=k B T Þ, with T being the temperature, k B the Boltzmann constant, and H the system Hamiltonian. The partition function encodes thermodynamics via the free-energy density f ¼ −ðk B T=NÞ log ½ZðTÞ , where N denotes the number of degrees of freedom.A phase transition, i.e., a sudden change of macroscopic behavior, is associated with a nonanalytical behavior of f as a function of temperature or another control parameter g such as an external magnetic field. QPTs [10] occur when T is kept at absolute 0, where the system's ground-state properties undergo a nonanalytic change as a function of g [see Fig. 1(a)]. Scenarios where boundary conditions are essential, such as for the Casimir effect, can be studied by boundary partition functions Z B ¼ hψ 0 je −RH jψ 0 i, where jψ 0 i encodes the spatial boundary conditions on two ends of the system at dis...