One of the central problems in the Banach space theory of the LP-spaces is to classify their complemented subspaces up to isomorphism (i.e., linear homeomorphism). Let us fix 1 < p < xc, p =# 2. There are five simple examples, LP, UP, 12, 12 @ Up, and (12 @ 12 @ ... )P. Although these were the only infinitedimensional ones known for some time, further impetus to their study was given by the discoveries of Lindenstrauss and Pelczyniski [15] and Lindenstrauss and Rosenthal [16]. These discoveries showed that a separable infinite-dimensional Banach space is isomorphic to a complemented subspace of LP if and only if it is isomorphic to 12 or is an EP-space, that is, equal to the closure of an increasing union of finite-dimensional spaces uniformly close to 1'P's. By making crucial use of statistical independence, the second author produced several more examples in [19], and the third author built infinitely many non-isomorphic examples in [23]. These discoveries left unanswered: Does there exist a Xp and infinitely many non-isomorphic Xp-complemented subspaces of LP (equivalently, are there infinitely many separable Ep A-spaces for some X depending on p)? We answer these questions by obtaining uncountably many non-isomorphic complemented subspaces of LP.* Before our work, it was suspected that every EP-space nonisomorphic to LP embedded in (12e 12e ... )P (for 2 < p < oc) (see Problem 1 of [23]). Indeed, all the known examples had this property. However our results show that there is no universal fp-space besides LP. To obtain these results, we use rather deep properties of martingales together with a new ordinal index, called the local LP-index, which assigns large countable ordinals to any