Recent in situ measurements by the MMS and Parker Solar Probe missions bring interest to small-scale plasma dynamics (waves, turbulence, magnetic reconnection) in regions where the electron thermal energy is smaller than the magnetic one. Examples of such regions are the Earth's magnetosheath and the vicinity of the solar corona, and they are also encountered in other astrophysical systems. In this brief review, we consider simple physical models describing plasma dynamics in such low-electron-beta regimes, discuss their conservation laws and their limits of applicability. Astrophysical plasmas (e.g., the Interstellar medium, solar wind, etc) are often in a state of a rough equipartition between the kinetic energies of the particles and the energy of the magnetic fields. However, there are important astrophysical and space environments, such as the Earth's magnetosphere and magnetosheath, and the solar corona and its vicinity, that are characterized by low electron plasma beta, that is, low ratio of electron thermal to magnetic energy, beta_e = 8 * pi * n_e * T_e / B^2 (e.g., Cranmer et al., 2009; Štverák et al., 2015; Bale et al., 2016; Chen et al., 2014), where n_e and T_e are the electron density and temperature, and B is the magnetic field strength. Such plasmas are also nearly collisionless in that the characteristic times of turbulent fluctuations are much shorter than the Coulomb collision times. The interest in plasma dynamics in low-beta regimes is also fueled by recent in situ measurements by NASA's MMS and Parker Solar Probe missions, as well as by the measurements expected from the Solar Orbiter spacecraft (e.g., Phan et al., 2018; Chen et al., 2020; Bale et al., 2019; Kasper et al., 2019). In this contribution we briefly review the theoretical frameworks for studying collisionless low-electron-beta plasma dynamics. In a weakly collisional plasma, the electrons and the ions do not exchange energy efficiently due to the strong difference in their masses. Therefore, it is a common situation that the ion temperature is different from the electron one. In our treatment of the problem we will, therefore, distinguish between the ion and electron betas beta_s = 8 * pi * n_s * T_s / B^2, where s = {e, i}. While we will concentrate on the case of small electron beta beta_e << 1, we will not necessarily assume the same for the ion beta, and will consider the cases of beta_i << 1 as well as beta_i ~ 1. For example, the Earth's magnetosphere is characterized by beta_i << m_e / m_i, beta_e << m_e / m_i, the solar corona and its vicinity correspond to beta_e <~ 0.01 and beta_i <~ 0.1, the Earth's magnetosheath beta_e ~ 0.1, beta_i <~ 1. Other environments with low electron beta include downstream regions of collisionless shocks and magnetospheres of accretion discs (e.g., Quataert, 1998; Vink et al., 2015; Ghavamian et al., 2013). The most rigorous treatment of a collisionless plasma is provided by the kinetic framework. However, kinetic framework presents considerable challenges for theoretical and especially numerical treatments. In many important cases, a simplified fluid-like description is possible that is much more physically transparent and allows for efficient numerical studies of plasma waves, turbulence, magnetic reconnection, structure formation, etc. The derivations of such simplified models can be performed using various approaches (reduced two-fluid, gyrofluid, gyrokinetic, kinetic, etc.), and such derivations are scattered in the literature. In this brief review, we discuss several models which we believe are relevant for the above mentioned space physics applications. Our goal is to present a unifying physical derivation of the governing equations, describe the corresponding conservation laws, and discuss the limits of applicability of each of the models. In this section we present a general derivation of the model equations, and then consider the limits of beta_i and beta_e mentioned in the introduction. As it is generally the case in magnetized plasma turbulence, we assume the presence of a uniform magnetic field (the guide field), which mimics the magnetic field of external sources or the magnetic field generated by large-scale turbulent motions. At small scales, the magnetic fluctuations are small, so we separate them from the guide field B = B_0 z_hat + delta B. We consider the case of small electron beta, so it will be easy to start with the equations describing the electron dynamics where we can neglect the effects related to the electron gyroradius. There are analytical and observational reasons to believe that small-scale fluctuations are oblique in that their wavenumbers along the guide field are much smaller than the wavenumbers in the perpendicular direction, k_parallel << k_perp. In the case of strong developed turbulence, critical balance is expressed by the ordering k_parallel / k_perp ~ |delta B| / B_0 ~ delta n / n_0 << 1. In a collisionless plasma, the electron gyro orbits drift in the field-perpendicular direction. To the zeroth and first orders in omega / Omega_e, this motion consists of the standard E x B drift and the polarization drift: v_perp = v_E - (m_e c / (e B^2)) * (d_E / dt) v_E, where v_E = (c / B^2) (E x B) and d_E / dt = d/dt + v_E . grad. Relating the parallel electric current to fluctuating fields from Ampere's law, neglecting the displacement current, and applying the electron continuity equation leads to: d/dt [ delta n_e / n_0 - delta B_z / B_0 + (m_e c^2 / (e B_0^2)) grad_perp^2 phi ] + (c / B_0) (z_hat x grad_perp phi) . grad_perp [ delta n_e / n_0 - delta B_z / B_0 + (m_e c^2 / (e B_0^2)) grad_perp^2 phi ] = - (c / (4 * pi * n_0 * e)) grad_parallel grad_perp^2 A_z. Combined with the ion counterpart and Gauss's law, the normalized charge continuity equation is obtained: d/dt grad_perp^2 phi + (z_hat x grad_perp phi) . grad_perp grad_perp^2 phi = - (v_A / (1 + v_A^2 / c^2)) grad_parallel grad_perp^2 A_z. Supplementing this with the parallel electron momentum equation: d/dt (1 - d_e^2 grad_perp^2) A_z + (z_hat x grad_perp phi) . grad_perp (1 - d_e^2 grad_perp^2) A_z = - v_A (d phi / dz) + (d_i / (n_0 m_i)) grad_parallel p_e. In the case of cold electrons and ions (phase velocity larger than electron thermal velocity, omega / k_parallel >> v_Te, implying beta_e << m_e/m_i and beta_i << 1), electron pressure in the momentum equation can be neglected: d/dt grad_perp^2 phi + (z_hat x grad_perp phi) . grad_perp grad_perp^2 phi = - v_A grad_parallel grad_perp^2 A_z d/dt (1 - d_e^2 grad_perp^2) A_z + (z_hat x grad_perp phi) . grad_perp (1 - d_e^2 grad_perp^2) A_z = - v_A (d phi / dz). The linear dispersion relation yields inertial Alfvén modes: omega^2 = (k_z^2 v_A^2) / (1 + k_perp^2 d_e^2). At scales k_perp d_e << 1, these turn into standard reduced MHD shear Alfvén modes. The electron inertial terms are relevant when beta_i << m_e / m_i. Conserved quantities in this regime are the total energy and generalized helicity: E = integral [ (grad_perp A_z)^2 + d_e^2 (grad_perp^2 A_z)^2 + (grad_perp phi)^2 ] d^3 x H = integral [ grad_perp^2 phi (1 - d_e^2 grad_perp^2) A_z ] d^3 x. In this regime, electrons are hot compared to the wave phase velocity (omega / k_parallel << v_Te), while ions remain cold. Expressing electron pressure as p_e = n_e T_e and applying the electron density relation yields: d/dt (delta n / n_0) + (z_hat x grad_perp phi) . grad_perp (delta n / n_0) = - d_i grad_parallel grad_perp^2 A_z. The linear dispersion relation becomes omega^2 = k_z^2 v_A^2 (1 + k_perp^2 rho_s^2) / (1 + k_perp^2 d_e^2), where rho_s = c_s / Omega_i is the ion-acoustic radius and c_s is the ion acoustic speed. The quadratic conservation laws are energy and generalized enstrophy: E = integral [ (grad_perp A_z)^2 + d_e^2 (grad_perp^2 A_z)^2 + (grad_perp phi)^2 + rho_s^2 (delta n / n_0)^2 ] d^3 x Omega_2 = integral [ delta n / n_0 - (1 / Omega_i) grad_perp^2 phi ]^2 d^3 x. When ions and electrons are both hot, the ion gyroscale is not small, and on sub-ion scales (k_perp^2 rho_i^2 >> 1) unmagnetized ions adjust according to the Boltzmann distribution delta n / n_0 = - e phi / T_i. The resulting closed system describes inertial kinetic-Alfvén waves: d/dt (1 + 2/beta_i - d_e^2 grad_perp^2) phi = v_A d_i^2 grad_parallel grad_perp^2 A_z d/dt (1 - d_e^2 grad_perp^2) A_z + (z_hat x grad_perp phi) . grad_perp (1 - d_e^2 grad_perp^2) A_z = - v_A (d phi / dz). The linear dispersion relation is omega^2 = (k_z^2 v_A^2 k_perp^2 d_i^2) / [ (1 + k_perp^2 d_e^2) (1 + 2/beta_i + k_perp^2 d_e^2) ]. The conserved quadratic invariants are energy and generalized helicity: E = integral [ phi (1 + 2/beta_i - d_e^2 grad_perp^2) phi - d_i^2 (grad_perp^2 A_z) (1 - d_e^2 grad_perp^2) A_z ] d^3 x H = integral [ (1 + 2/beta_i - d_e^2 grad_perp^2) phi (1 - d_e^2 grad_perp^2) A_z ] d^3 x. Several physical models of nonlinear plasma dynamics at low electron beta have been described, which are relevant for space physics applications ranging from the Earth's magnetosphere to the magnetosheath to the solar corona. These models provide insights into turbulent cascades, magnetic reconnection, and wave phenomena. While fluid-like models do not self-consistently incorporate kinetic dissipation like Landau damping, the critical balance condition implies dissipative kinetic terms may be incorporated as linear terms without fundamentally altering the nonlinear dynamics captured by these simplified frameworks. Authors: Stanislav Boldyrev (Department of Physics, University of Wisconsin-Madison; Center for Space Plasma Physics, Space Science Institute), Nuno F. Loureiro (Plasma Science and Fusion Center, Massachusetts Institute of Technology), Vadim Roytershteyn (Center for Space Plasma Physics, Space Science Institute). Funding: SB was supported by NSF (PHY-1707272, PHY-2010098), NASA (80NSSC18K0646), and DOE (DE-SC0018266). NFL was funded by NSF CAREER Award No. 1654168 and NSF-DOE Partnership in Basic Plasma Science and Engineering (PHY-2010136). VR was supported by DOE Grant No. DE-SC0019315.