A systematic and unified approach to transformations and symmetries of a class of variable coefficient heat type linear partial differential equations (PDEs) is presented. The complete symmetry group classification is re-performed in the current context. A useful criterion which is necessary and sufficient for being reducible 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. Multidimensional parabolic PDEs of heat and Schrodinger type are also considered. Published by AIP Publishing.