  A systematic and unified approach to transformations and symmetries of
general second order linear parabolic partial differential equations is
presented. Equivalence group is used to derive the Appell type transformations,
specifically Mehler's kernel in any dimension. The complete symmetry group
classification is re-performed. A new criterion which is necessary and
sufficient for reduction to the standard heat equation by point transformations
is established. A similar criterion is also valid for the equations to have a
four- or six-dimensional symmetry group (nontrivial symmetry groups). In this
situation, the basis elements are listed in terms of coefficients. A number of
illustrative examples are given. In particular, some applications from the
recent literature are re-examined in our new approach. Applications include a
comparative discussion of heat kernels based on group-invariant solutions and
the idea of connecting Lie symmetries and classical integral transforms
introduced by Craddock and his coworkers. Multidimensional parabolic PDEs of
heat and Schr\"odinger type are also considered.
