Two classes of processes are considered. One is a class of spectrally positive infinitely divisible processes which includes all such stable processes. The other is a class of processes constructed from the sequence of partial sums of independent identically distributed positive random variables. A condition analogous to regular variation of the tails is imposed. Then a large deviation principle and a Strassen-type law of the iterated logarithm are presented. These theorems focus on unusually large values of the processes. They are expressed in terms of Skorokhod's M1 topology. 1. Introduction. The purpose of this paper is to present a large deviation principle (LDP) and an analogue of Strassen's law of the iterated logarithm (LIL) for two classes of stochastic processes. The LDP and LIL provide information about the likelihood of unusually large values of the processes. The first class of processes consists of certain spectrally positive infinitely divisible (inf. div.) processes, including all spectrally positive stable processes. The second is a class of processes obtained from sequences of partial sums (p. sums) of independent identically distributed positive random variables. The results and the proofs for the two types of processes are very similar, so it is efficient to consider them together. Likewise, there are close ties between the proofs of the LDP and the LIL; probability estimates for the former are used in conjunction with the Borel-Cantelli lemma to prove the latter. The inf. div. processes and p. sums processes considered here are represented as integrals with respect to planar point processes. Such representations permit us to apply limit theorems proved for the point processes in O'Brien and Vervaat (1996). These theorems are summarized in Section 3. If the representations were continuous, the results of the present paper would be trivial. As it is, we must work hard to reduce the problems to the point process results. We now describe the two classes of processes, beginning with the inf. div. case. Let v be a measure on (0, oo] with v({oo}) = 0 and 0 0. Some further restrictions, including an analogue of regular variation, will be imposed on v later. Let E := [0, co) x (0, oc]. We generally use the symbol t for the first (horizontal or time) coordinate of E and x or y for the second. Next, let F be the Poisson point process (random measure) on E with intensity dtv(dx).