  Vector spherical harmonics on the unit sphere of $\mathbb{R}^3$ have broad
applications in geophysics, quantum mechanics and astrophysics. In the
representation of a tangent vector field, one needs to evaluate the expansion
and the Fourier coefficients of vector spherical harmonics. In this paper, we
develop fast algorithms (FaVeST) for vector spherical harmonic transforms on
these evaluations. The forward FaVeST evaluates the Fourier coefficients and
has a computational cost proportional to $N\log \sqrt{N}$ for $N$ number of
evaluation points. The adjoint FaVeST which evaluates a linear combination of
vector spherical harmonics with a degree up to $\sqrt{M}$ for $M$ evaluation
points has cost proportional to $M\log\sqrt{M}$. Numerical examples of
simulated tangent fields illustrate the accuracy, efficiency and stability of
FaVeST.
