Axios does not support canceling requests at the moment. Please see this issue for details.

UPDATE: Cancellation support was added in axios v0.15.

EDIT: The axios cancel token API is based on the withdrawn cancelable promises proposal.

UPDATE 2022: Starting from v0.22.0 Axios supports AbortController to cancel requests in fetch API way:

Example:

const controller = new AbortController();

axios.get('/foo/bar', {
   signal: controller.signal
}).then(function(response) {
   //...
});
// cancel the request
controller.abort()

Since axios v0.15 you can cancel request:

You can also create a cancel token by passing an executor function to the CancelToken constructor:

var CancelToken = axios.CancelToken;
var cancel;

axios.get('/user/12345', {
  cancelToken: new CancelToken(function executor(c) {
    // An executor function receives a cancel function as a parameter
    cancel = c;
  })
});

// cancel the request
cancel();

For more information please have a look Cancellation. And according to your code:

import React from 'react'
import axios from 'axios';

export function autocompleteSearchTest(value) {
    if (cancel != undefined) {
        cancel();
    }
    return axios.get(`https://jqueryui.com/resources/demos/autocomplete/search.php`, {
        cancelToken: new CancelToken(function executor(c) {
            // An executor function receives a cancel function as a parameter
            cancel = c;
        }),
        params: {
            q: value
        }
    })
        .then(response => {
            return response.data.response;
        })
        .catch(error => {
            const result = error.response;
            return Promise.reject(result);
        });
}

var CancelToken = axios.CancelToken;
var cancel;

export class AutoComplete extends React.Component {

    constructor() {
        super()
        this.state = {
            search: ''
        };
        this.handleSearchChange = this.handleSearchChange.bind(this);
    }

    handleSearchChange(event) {
        const target = event.target;
        const value = target.value;
        this.setState({
            search: value
        });
        autocompleteSearchTest(value)
    }

    render() {
        return (
            <div className="App-intro Dialog-section">
                <h2>AutoComplete</h2>
                <div className="form-control">
                    <label htmlFor="password">Lookup :</label>
                    <input name="search" type="text" value={this.state.search}
                           onChange={this.handleSearchChange}
                           id="password" ref="password" placeholder="Enter line"/>
                </div>
            </div>
        )
    }
}

export default AutoComplete;

And here it's