We study the two-dimensional stochastic nonlinear heat equation (SNLH) and stochastic damped nonlinear wave equation (SdNLW) with an exponential nonlinearity lambda beta e^{beta u }, forced by an additive space-time white noise. (i) We first study SNLH for general lambda in {mathbb {R}}. By establishing higher moment bounds of the relevant Gaussian multiplicative chaos and exploiting the positivity of the Gaussian multiplicative chaos, we prove local well-posedness of SNLH for the range 0< beta ^2 < frac{8 pi }{3 + 2 sqrt{2}} simeq 1.37 pi . Our argument yields stability under the noise perturbation, thus improving Garban’s local well-posedness result (2020). (ii) In the defocusing case lambda >0, we exploit a certain sign-definite structure in the equation and the positivity of the Gaussian multiplicative chaos. This allows us to prove global well-posedness of SNLH for the range: 0< beta ^2 < 4pi . (iii) As for SdNLW in the defocusing case lambda > 0, we go beyond the Da Prato-Debussche argument and introduce a decomposition of the nonlinear component, allowing us to recover a sign-definite structure for a rough part of the unknown, while the other part enjoys a stronger smoothing property. As a result, we reduce SdNLW into a system of equations (as in the paracontrolled approach for the dynamical Phi ^4_3-model) and prove local well-posedness of SdNLW for the range: 0< beta ^2 < frac{32 - 16sqrt{3}}{5}pi simeq 0.86pi . This result (translated to the context of random data well-posedness for the deterministic nonlinear wave equation with an exponential nonlinearity) solves an open question posed by Sun and Tzvetkov (2020). (iv) When lambda > 0, these models formally preserve the associated Gibbs measures with the exponential nonlinearity. Under the same assumption on beta as in (ii) and (iii) above, we prove almost sure global well-posedness (in particular for SdNLW) and invariance of the Gibbs measures in both the parabolic and hyperbolic settings. (v) In Appendix, we present an argument for proving local well-posedness of SNLH for general lambda in {mathbb {R}}without using the positivity of the Gaussian multiplicative chaos. This proves local well-posedness of SNLH for the range 0< beta ^2 < frac{4}{3} pi simeq 1.33 pi , slightly smaller than that in (i), but provides Lipschitz continuity of the solution map in initial data as well as the noise.