  Layered materials with robust magnetic ordering have been attracting
significant research interest. In recent experiments, a new layered material
1$T$-CrTe$_{2}$ has been synthesized and exhibits ferromagnetism above the room
temperature. Here, based on first-principles calculations, we investigate the
electronic, magnetic, and transport properties of 1$T$-CrTe$_{2}$, both in the
bulk and in the two-dimensional (2D) limit. We show that 1$T$-CrTe$_{2}$ can be
stable in the monolayer form, and has a low exfoliation energy. The monolayer
structure is an intrinsic ferromagnetic metal, which maintains a high Curie
temperature above the room temperature. Particularly, we reveal interesting
features in the anomalous Hall transport. We show that in the ground state,
both bulk and monolayer 1$T$-CrTe$_{2}$ possess vanishing anomalous Hall
effect, because the magnetization preserves one vertical mirror symmetry. The
anomalous Hall conductivity can be made sizable by tuning the magnetization
direction or by uniaxial strains that break the mirror symmetry. The
room-temperature 2D ferromagnetism and the tunable anomalous Hall effect make
the material a promising platform for nanoscale device applications.
