The Kardar-Parisi-Zhang (KPZ) equation is a singular stochastic partial differential equation which describes the random interface growth in a universal way. Indeed, the KPZ equation has been derived from various types of microscopic systems through a scaling procedure, which phenomenon is referred to as the weak KPZ universality. In this talk, I will introduce two typical regimes from which the KPZ equation is derived in the limit: the weakly asymmetric regime, and the strongly asymmetric regime with the high-temperature limit. After showing some recent results in each of these regimes, I will show some conjecture which enables us to obtain a comprehensive description of the weak KPZ universality in interacting particle systems.