We develop a multivalued theory for the stability operator of (a constant multiple of) a minimally immersed submanifold Σ of a Riemannian manifold M. We define the multiple valued counterpart of the classical Jacobi fields as the minimizers of the second variation functional defined on a Sobolev space of multiple valued sections of the normal bundle of Σ in M, and we study existence and regularity of such minimizers. Finally, we prove that any Q-valued Jacobi field can be written as the superposition of Q classical Jacobi fields everywhere except for a relatively closed singular set having codimension at least two in the domain. Assumption 1.1. We will consider:(M) a closed (i.e. compact with empty boundary) Riemannian manifold M of dimension m + k and class C 3,β for some β ∈ (0, 1); (S) a compact oriented minimal submanifold Σ of the ambient manifold M of dimension dim(Σ) = m and class C 3,β . Without loss of generality, we will also regard M as an isometrically embedded submanifold of some Euclidean space R d . We will let n := d − m and K := d − (m + k) be the codimensions of Σ and M in R d respectively.Let Σ m → M m+k ⊂ R d be as in Assumption 1.1. The Euclidean scalar product in R d is denoted ·, · . The metric on M and Σ is induced by the flat metric in R d : therefore, the same symbol will also denote the scalar product between tangent vectors to M or to Σ.The tangent space to M at a point z will be denoted T z M. The maps p M z : R d → T z M and p M⊥ z : R d → T ⊥ z M denote orthogonal projections of R d onto the tangent space to M at