Edit

The below solution catches only render errors because they are sync in nature. That is how JS works, it has nothing to do with React, which is what the OP asked about.

Original post

React 16 introduced Error Boundaries and the componentDidCatch lifecycle method:

class ErrorBoundary extends React.Component {
  constructor(props) {
    super(props);
    this.state = { hasError: false };
  }

  componentDidCatch(error, info) {
    // Display fallback UI
    this.setState({ hasError: true });
    // You can also log the error to an error reporting service
    logErrorToMyService(error, info);
  }

  render() {
    if (this.state.hasError) {
      // You can render any custom fallback UI
      return <h1>Something went wrong.</h1>;
    }
    return this.props.children;
  }
}

Then you can use it as a regular component:

<ErrorBoundary>
  <MyWidget />
</ErrorBoundary>

Or you can wrap your root component with the npm package react-error-boundary, and set a fallback component and behavior.

import {ErrorBoundary} from 'react-error-boundary';

const myErrorHandler = (error: Error, componentStack: string) => {
  // ...
};

<ErrorBoundary onError={myErrorHandler}>
  <ComponentThatMayError />
</ErrorBoundary>