1. Introduction. The present note contains proofs of uniqueness theorems for the ordinary differential equation y'=f (x, y), and for the hyperbolic partial differential equation u, =f(x, y, u, ux, ui,), under what may be called Nagumo uniqueness conditions. Reference is made to Kamke [5] for description and literature concerning the uniqueness condition on f(x, y) (which is less restrictive than the Lipschitz condition) which was introduced into the theory of the ordinary differential equation y'=f(x, y) by Nagumo [1]. In connection with the partial differential equation uy =f(x, y, u, UZI un), uniqueness conditions more general than the Lipschitz condition were introduced by Walter [7]. ?2 deals with the ordinary differential equation, first under the assumption of a Lipschitz condition, and then a Nagumo condition. The argument in the case of the Lipschitz condition is included because it appears to be (under the assumptions used) simpler and more direct than that currently employed in textbooks, which usually rely on the theory of the definite integral to some extent (see, however, Caratheodory [2], who treats a more general case). The argument in the case of the Nagumo condition is also independent of the theory of the definite integral, and is to be compared with the proofs in Kamke [5] and Perron [4]. ?3 contains an extension of a Nagumo type theorem of Walter [7], for the characteristic boundary value