We introduce and study the notion of non-interactive secure multiparty computation (NIMPC). An NIMPC protocol for a function f(x1,...,x n) is specified by a joint probability distribution R = (R 1,...,Rn) and local encoding functions Enc i(xi,ri), 1 ≤ i ≤ n. Given correlated randomness (r1,...,rn) ∈R R, each party Pi, using its input xi and its randomness ri, computes the message mi = Enci(xi, r i). The messages m1,...,mn can be used to decode f(x1,...,xn). For a set T ⊆ [n], the protocol is said to be T-robust if revealing the messages (Enci(xi, ri))i∈T together with the randomness (r i)i∈T gives the same information about (x 1i∈T as an oracle access to the function f restricted to these input values. Namely, a coalition T can learn no more than the restriction of f fixing the inputs of uncorrupted parties, which, in this non-interactive setting, one cannot hope to hide. For 0 ≤ t ≤ n, the protocol is t-robust if it is T-robust for every T of size at most t and it is fully robust if it is n-robust. A 0-robust NIMPC protocol for f coincides with a protocol in the private simultaneous messages model of Feige et al. (STOC 1994). In the setting of computational (indistinguishability-based) security, fully robust NIMPC is implied by multi-input functional encryption, a notion that was recently introduced by Goldwasser et al. (Eurocrypt 2014) and realized using indistinguishability obfuscation. We consider NIMPC in the information-theoretic setting and obtain unconditional positive results for some special cases of interest: - Group products. For every (possibly non-abelian) finite group G, the iterated group product function f(x1,...,x n) = x1x2...xn admits an efficient, fully robust NIMPC protocol. - Small functions. Every function f admits a fully robust NIMPC protocol whose complexity is polynomial in the size of the input domain (i.e., exponential in the total bit-length of the inputs). - Symmetric functions. Every symmetric function f:Xn → Y, where X is an input domain of constant size, admits a t-robust NIMPC protocol of complexity nO(t). For the case where f is a w-out-of-n threshold function, we get a fully robust protocol of complexity nO(w). On the negative side, we show that natural attempts to realize NIMPC using private simultaneous messages protocols and garbling schemes from the literature fail to achieve even 1-robustness.