Wyss was born in Bern, Switzerland in 1935. His Father sold mechanical calculators and his mother was a homemaker. He was raised in an apartment with two sisters. After receiving a master's degree in Civil and Structural Engineering from the Swiss Federal Institute of Technology Zurich in 1959, Wyss earned an MBA from the Harvard University Graduate School of Business in 1965. Following that, he worked in various positions in the textile industry, including plant Engineer and project manager for Chrysler in Pakistan, Turkey, and the Philippines.

Wyss has stated that he became passionate about the American West and land preservation after visiting the U.S. in 1958 as a student and taking a summer job as a surveyor with the Colorado Highway Department. In 1998, he created the Wyss Foundation to establish and sponsor informal partnerships between non-governmental organizations and the United States government to place large swathes of land under permanent protection in the American West. By 2006, via the initial efforts of the foundation, almost 4,400,000 acres (18,000 km) of public land had been labeled as national monuments and national conservation areas. The organization also sponsors The Wyss Scholars Program for graduate-level education in conservation.

In 1977, Wyss founded and became President of Synthes USA, the U.S. division of Switzerland-based Synthes medical device manufacturer making internal screws and plates for broken bones. In an early initiative, Wyss opened a Synthes USA Manufacturing plant in Colorado. Prior to that, another Switzerland company manufactured Synthes' devices and exported them to the U.S. Under Wyss' control, the U.S. division expanded its sales team and trained Surgeons how to use its products. Wyss served as Synthes' worldwide CEO and chairman until his resignation as CEO in 2007. He maintained his post as company chairman until Johnson & Johnson acquired Synthes in 2012. During his tenure, Wyss said discussions of new products made up one-third of board meetings. A manager assigned to the Norian project testified before a grand jury that “for somebody who is at his level and his level of success, I would say he [Wyss] has a surprising amount of contact with what's going on.” Staff recalled meetings in which he intensively probed their projects.

Wyss is an active hiker, skier and backpacker. He is also a hobby pilot. He lives in Wyoming, where he is involved in outdoor education programs and funds local efforts to conserve wildlife habitat and public lands in the Rocky Mountains. In 2000, Wyss purchased the 900-acre (3.6 km) Halter Ranch & Vineyard in western Paso Robles, California. The ranch includes an 1,800 acre wildlife preserve and a 281 acre vineyard producing 13 varietals using methods that are "Sustainability in Practice" certified. The ranch hosts tours and was named "Best Vineyard Experience" by Sunset Magazine in 2015.

According to Forbes, Wyss is "among the most philanthropic people in the world". Between 2004 and 2008, Businessweek estimated that Wyss personally donated nearly USD$277 million. His giving has increased since the sale of Synthes in 2012 and in 2013 he signed The Giving Pledge, agreeing to give away the majority of his fortune. The assets of his charitable foundations equal nearly $2 billion.

In 2007, he received the Harvard Business School Alumni Achievement Award, and in fall 2008, it was announced that Wyss donated the largest single endowment from one source in Harvard's history when he gave $125 million to found a multidisciplinary institute, the Wyss Institute for Biologically Inspired Engineering at Harvard University.

In 2009, top executives at Synthes were indicted by U.S. Attorneys for Eastern Pennsylvania for using an untested calcium-phosphate-based bone cement on human patients without the authorization from the Food and Drug Administration, which resulted in the deaths of three people. Wyss was not indicted but four of Synthes' top executives were convicted and sentenced to prison terms.

In 2010, Wyss personally gave The Nature Conservancy $35 million to purchase 310,000 acres in Montana as part of one of the largest private conservation purchases in the United States. He donated $4.25 million to The Trust for Public Land in 2013 for the purchase of oil and gas leases in Wyoming to prevent development in the Hoback Basin. In 2016, Wyss made another donation to the Trust for Public Land that resulted in the expansion of Saguaro National Park in Arizona by 300 acres, including a mile and a half of Rincon Creek.

Wyss is involved with The Wilderness Society, Rails-to-Trails, and serves on the boards of the Southern Utah Wilderness Alliance, Center for American Progress, and the Grand Canyon Trust. In 2011, Wyss won the Robert Marshall Award from The Wilderness Society for his conservation work.

In 2012-13, he announced the creation of the Campus Biotech and of its Wyss Center for Bio- and Neuro-engineering in Geneva, Switzerland. In 2014, Wyss donated $120 million to the Swiss Federal Institute of Technology Zurich and the University of Zurich for the new Wyss Translational Center Zurich.

In 2015, Wyss publicly declared himself to be in favour of higher inheritance taxes (estate/death duty taxes) for the wealthy in Switzerland.

As of 2017, Wyss ranks 281 on the Forbes list of billionaires, with an estimated net worth of approximately $5.5 billion. He ranks number 235 on the Bloomberg list of billionaires.