The Nonlinear Noisy Leaky Integrate and Fire (NNLIF) models are one of the simplest mean-field models used to describe the behaviour of large groups of neurons at the level of their membrane potential. In recent years, NNLIF models have been studied from a mathematical point of view; at the microscopic level, using Stochastic Differential Equations (SDE) and at mesoscopic/macroscopic level, through the nonlinear Fokker-Planck type equations (FPE). The considerable amount of publications and unanswered questions on these models reveal their high mathematical complexity, despite their simplicity. In this talk we focus on analysing the asymptotic behaviour of this type of models in terms of the values of the connectivity parameter and of the transmission delay. These models exhibit a wide variety of phenomena: stable/unstable steady states, blow-up/global existence, and periodic solutions. Through a numerical study of their particle systems, we answer the question of what happens to the system when all neurons fire at the same time. Using spectral gap techniques and entropy method in kinetic theory, and tools from delay equations we are able to give a simple criterion for the stability and instability of equilibria. Finally, we discuss the presence of periodic solutions for inhibitory networks with high transmission delay. This talk is based on works in collaboration with Cañizo and Ramos-Lora and previous works in collaboration with Carrillo, Perthame, Roux, Schneider and Salort.