Authorâ€™s abstract: We are dealing with the concept of

$d$-dimensional orthogonal (abbreviated

$d$-orthogonal) polynomials, that is to say polynomials verifying one standard recurrence relation of order

$d+1$. Among the

$d$-orthogonal polynomials one singles out the natural generalizations of certain classical orthogonal polynomials. In particular, we are concerned, in the present paper, with the solution of the following problem (P): Find all polynomial sequences which are at the same time Appell polynomials and

$d$-orthogonal. The resulting polynomials are a natural extension of the Hermite polynomials. A sequence of these polynomials is obtained. All the elements of its

$(d+1)$-order recurrence are explicitly determined. A generating function, a

$(d+1)$-order differential equation satisfied by each polynomial and a characterization of this sequence through a vectorial functional equation are also given. Among such polynomials one singles out the

$d$-symmetrical ones (Definition 1.7) which are the

$d$-orthogonal polynomials analogous to the Hermite classical ones. When

$d=1$ (ordinary orthogonality), we meet again the classical orthogonal polynomials of Hermite.