As the year began, Amgen Inc.’s Chief Executive Kevin Sharer boasted his biotechnology company had just finished “one of our very, very best years ever.”

The world’s largest biotech company by sales enjoyed a market capitalization above $80 billion, higher than many big pharmaceutical companies. Sales were rising at a double-digit clip.

But today Amgen and Mr. Sharer are on the defensive. Safety questions have emerged on two of its biggest drugs, which treat anemia in cancer and kidney-failure patients. A black-box warning placed March 9 on the labels of those and other anemia drugs urges doctors to use the smallest possible dose of the drugs. Amgen stock has plunged, wiping out roughly $20 billion in market value since its 52-week high in October.

Amgen has arrived as a big pharmaceutical company — and now confronts some of the same problems as Pfizer Inc. or Merck & Co. Among them: heavy reliance on a few blockbusters, an uncertain pipeline of new drugs despite heavy research spending, questions about safety and marketing and, recently, the prospect of competition from generics makers.

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