Single-photon emitters are essential for enabling several emerging applications in quantum information technology, quantum sensing and quantum communication. Scalable photonic platforms capable of hosting intrinsic or directly embedded sources of single-photon emission are of particular interest for the realization of integrated quantum photonic circuits. Here, we report on the first-time observation of room-temperature single-photon emitters in silicon nitride (SiN) films grown on silicon dioxide substrates. As SiN has recently emerged as one of the most promising materials for integrated quantum photonics, the proposed platform is suitable for scalable fabrication of quantum on-chip devices. Photophysical analysis reveals bright (>$10^5$ counts/s), stable, linearly polarized, and pure quantum emitters in SiN films with the value of the second-order autocorrelation function at zero time delay $g^{(2)}(0)$ below 0.2 at room temperatures. The emission is suggested to originate from a specific defect center in silicon nitride due to the narrow wavelength distribution of the observed luminescence peak. Single-photon emitters in silicon nitride have the potential to enable direct, scalable and low-loss integration of quantum light sources with the well-established photonic on-chip platform.