  This article presents a review of the finite element method (FEM) model based
on the $H$ formulation of Maxwell's equations used to calculate AC losses in
high temperature superconductor (HTS) tapes, cables and windings for different
applications. This model, which uses the components of the magnetic field as
state variables, has been gaining a great popularity and has been in use in
tens of research groups around the world. This contribution first reviews the
equations on which the model is based and their implementation in finite
element method programs for different cases, such 2D longitudinal and
axis-symmetric geometries, 3D geometries. Modeling strategies to tackle large
number of HTS tapes, such as multi-scale and homogenization methods, are also
introduced. Then, the second part of the article reviews the applications for
which the $H$ formulations has been used to calculate AC losses, ranging from
individual tapes, to complex cables and large magnet windings. Afterwards, a
section is dedicated to the discussion of the $H$ formulation in terms of
computational efficiency and easiness of implementation. Its pros and cons are
listed. Finally, the last section draws the main conclusions.
