Dr. Leonard Peikoff was a long time associate of Ayn Rand and was named by her as her legal heir. He has written and lectured widely on the principles of Ayn Rand’s philosophy and their application. He currently, among other endeavors, alternates with Yaron Brook in answering questions on the application of Objectivism via a weekly podcast.

From the biography on his site:

“I had the extraordinary good fortune to read Atlas Shrugged in manuscript as it was being written, and to ask the author all the questions I wished about her ideas,” Peikoff says. “My knowledge of philosophy was primitive at the time, but she was tremendously gifted at explaining her ideas. She was also patient and recommended many books for me to read. We talked philosophy late into the night on countless occasions. It was, for me, an invaluable education.”
Like several other bright young men who were part of the Rand inner circle (among them Alan Greenspan, former Chairman of the Federal Reserve Board), Peikoff was attracted to Objectivism because of its view “of man as a heroic being, with his own happiness as the moral purpose of his life, with productive achievement as his noblest activity, and reason as his only absolute.” From 1957 until 1973, Peikoff taught philosophy at Hunter College, Long Island University, New York University, the University of Denver and the Polytechnic Institute of Brooklyn.

• America vs. Americans – Leonard Peikoff - Leonard Peikoff
  
• Can Capitalism Survive? - Leonard Peikoff
  

  In this 1990 lecture, philosopher Leonard Peikoff offers a business-savvy audience his perspective on whether free market capitalism can survive. Peikoff identifies three pillars of capitalism, examining each one’s status in the present day. He also also leads the audience through a quick sweep of American history, commenting on the decline in freedom over time.
• Introduction to Logic - Leonard Peikoff
  

  This course (with exercises) covers the standard topics taught in introductory courses in Aristotelian logic. It defines the principles of valid reasoning, and discusses prevalent logical fallacies. It formalizes the steps by which one derives conclusions from premises, and it provides a methodology by which to evaluate one's own thinking processes. (Each lecture includes a question period.)
• Leonard Peikoff - Leonard Peikoff
  

  This is the web home of Dr. Leonard Peikoff, long time friend of Ayn Rand and the leading expert on the philosophy of Objectivism. On the website you can find the archive of his weekly podcast, which is currently alternates between episodes by him and episodes by Yaron Brook. Also on the site are links to his numerous lectures, essays and articles.
• Objective Communication - Leonard Peikoff
  

  Ayn Rand’s philosophy of Objectivism is increasingly influencing the shape of the world from business and politics to achieving personal goals. Here, Leonard Peikoff—Rand’s heir—explains how you can communicate philosophical ideas with conviction, logic, and, most of all, reason.
• Objective Communication - Leonard Peikoff
  

  In this 1980 ten-lecture course, Leonard Peikoff identifies principles for the effective communication of ideas, applying them to three areas: writing, speaking and arguing. This course is concerned, not with the style of a presentation, but with its substance, i.e., with the basic methods necessary to achieve a clear, absorbing presentation of your viewpoint.

  Peikoff draws on principles from such diverse fields as epistemology, drama, education and polemics. Throughout the sessions, students give brief presentations, which Peikoff critiques based on the principles he has identified. The subjects of these presentations are related to aspects of Objectivism, so listeners of this course also have the opportunity to expand or refresh their knowledge of Ayn Rand’s philosophy.

• Objectivism: The Philosophy of Ayn Rand - Leonard Peikoff
  

  ometimes referred to as OPAR, this is a great one volume summary or overview of Objectivism, covering everything from metaphysics, the fundamental nature of reality, to aesthetics, the nature and understanding of art.
• Principles of Grammar - Leonard Peikoff
  

  The science that studies the methods of combining concepts i.e.,words into sentences is grammar. Discover how the normally dry subject of grammar can be transformed into an engrossing, epistemological field of study. Learn why a mastery of the principles of grammar is essential for precision in thinking and writing. This is grammar as it should be but never is taught.
• Teaching Johnny to Think - Leonard Peikoff, Marlene Trollope
  

  Dr. Peikoff makes a compelling case for a rational system of education by contrasting three schools of philosophy and the different educational alternatives they propose to replace our present system. He translates the usual abstract discussions in this field into material easily comprehensible to the reader. In the process, he defines a proper methodology and curriculum that will produce thinking high school graduates confident of their ability to achieve their goals.
• The Art of Thinking - Leonard Peikoff
  

  Want to Improve Your Method of Thinking? This is a course on what to do with your mind during the act of thought, when to do it and how to do it. Dr. Peikoff teaches you how to make the principles of Objectivist epistemology the guide of your own daily thought processes. These lectures are part new theory and part exercises.
• The Sanction of the Victim (Leonard Peikoff presentation) - Leonard Peikoff
  

  The talk Ayn Rand was to deliver at Ford Hall Forum in 1982, presented movingly after her death, and in tribute to her, by Leonard Peikoff. (Includes his recollections of Ayn Rand's final weeks, and his views on Objectivism's future.)