Contemporary peer code review is a lightweight, asynchronous method for ensuring high-quality code. While this practice has been shown to be beneficial to help developers identify and remove faults from code, it is underutilized in scientific software. To remedy this situation, we propose a hands-on tutorial. The first part of the tutorial will provide important background information and motivation for the usefulness of contemporary peer code review. During this portion of the tutorial, we will describe the peer code review process, introduce the available support tools, and describe some of our own research results on peer code review in scientific software, open-source software, and commercial software. The second part of the tutorial will be more hands-on. We will first teach attendees how to use the different code review support tools commonly used in open-source and commercial software, e.g. Gerrit, Review Board, Phabricator, and Crucible. We will then provide the attendees with code review exercise in which they can use one or more tools and gain some experience. We will conclude the tutorial with a debrief of the hands-on exercise, a discussion of peer code review needs in scientific software, and an overview of peer code review best practices.