The problem of low-frequency current noise with a $1/f$ spectrum in solids (flicker noise) is reviewed. The fundamental general properties of this noise are described: frequency-dependence of the spectral density, time-dependence of the correlation function, form of the distribution function of the fluctuations, dependence of the spectral density on the potential applied to the specimen, anisotropy of the fluctuations of the resistivity tensor, correlation length of the fluctuations, dependence of the noise on the dimensions of the specimen and on the concentration of current carriers, the empirical Hooge relationship, etc. A model of $1/f$ noise is presented and discussed that associates its spectrum with the presence in solids of an extensive hierarchy of relaxation times. Concrete models of systems having an exponentially broad spectrum of relaxation times are described (the McWhorter model, two-level tunneling systems, the disordered Ising kinetic model). The theory and experimental data on current noise caused by temperature fluctuations are analyzed. The problem is treated of surface noise in semiconductors and $1/f$ noise in metal films.