my solution

src/App.tsx

@@ -1,6 +1,5 @@
-import { PeoplePage } from './components/PeoplePage';
+import { Outlet } from 'react-router-dom';
 import { Navbar } from './components/Navbar';
-
 import './App.scss';
 
 export const App = () => {
@@ -10,9 +9,7 @@ export const App = () => {
 
       <div className="section">
         <div className="container">
-          <h1 className="title">Home Page</h1>
-          <h1 className="title">Page not found</h1>
-          <PeoplePage />
+          <Outlet />
         </div>
       </div>
     </div>

src/Root.tsx

@@ -0,0 +1,32 @@
+import {
+  Navigate,
+  Route,
+  HashRouter as Router,
+  Routes,
+} from 'react-router-dom';
+import { App } from './App';
+import { HomePage } from './components/HomePage/HomePage';
+import { PeoplePage } from './components/PeoplePage';
+
+export const Root = () => (
+  <Router>
+    <Routes>
+      <Route path="/" element={<App />}>
+        <Route index element={<HomePage />} />
+        <Route path="people" element={<PeoplePage />} />
+        <Route path="home" element={<Navigate to="/" replace />} />
+        <Route path="people/:slug" element={<PeoplePage />} />
+        <Route
+          path="*"
+          element={
+            <main className="section">
+              <div className="container">
+                <h1 className="title">Page not found</h1>
+              </div>
+            </main>
+          }
+        />
+      </Route>
+    </Routes>
+  </Router>
+);

src/components/HomePage/HomePage.tsx

@@ -0,0 +1,9 @@
+import React from 'react';
+
+export const HomePage: React.FC = () => (
+  <main className="section">
+    <div className="container">
+      <h1 className="title">Home Page</h1>
+    </div>
+  </main>
+);

src/components/Navbar.tsx

@@ -1,26 +1,41 @@
-export const Navbar = () => {
+import React from 'react';
+import cn from 'classnames';
+import { Link, useLocation } from 'react-router-dom';
+
+export const Navbar: React.FC = () => {
+  const location = useLocation();
+
   return (
-    <nav
-      data-cy="nav"
-      className="navbar is-fixed-top has-shadow"
-      role="navigation"
-      aria-label="main navigation"
-    >
-      <div className="container">
-        <div className="navbar-brand">
-          <a className="navbar-item" href="#/">
-            Home
-          </a>
+    <>
+      <nav
+        data-cy="nav"
+        className="navbar is-fixed-top has-shadow"
+        role="navigation"
+        aria-label="main navigation"
+      >
+        <div className="container">
+          <div className="navbar-brand">
+            <Link
+              className={cn('navbar-item', {
+                'has-background-grey-lighter': location.pathname === '/',
+              })}
+              to="/"
+            >
+              Home
+            </Link>
 
-          <a
-            aria-current="page"
-            className="navbar-item has-background-grey-lighter"
-            href="#/people"
-          >
-            People
-          </a>
+            <Link
+              className={cn('navbar-item', {
+                'has-background-grey-lighter':
+                  location.pathname.startsWith('/people'),
+              })}
+              to="/people"
+            >
+              People
+            </Link>
+          </div>
         </div>
-      </div>
-    </nav>
+      </nav>
+    </>
   );
 };

src/components/PeopleFilters.tsx

@@ -1,16 +1,105 @@
+import { useSearchParams } from 'react-router-dom';
+import { SexEnum } from '../types/enums';
+import cn from 'classnames';
+
 export const PeopleFilters = () => {
+  const [searchParams, setSearchParams] = useSearchParams();
+  const query = searchParams.get('query') || '';
+  const centuries = searchParams.getAll('century') || [];
+  const allCenturies = [16, 17, 18, 19, 20];
+
+  function handleSexFiltering(
+    event: React.MouseEvent<HTMLAnchorElement>,
+    sex: string | null,
+  ) {
+    event.preventDefault();
+
+    const params = new URLSearchParams(searchParams);
+
+    if (sex) {
+      params.set('sex', sex);
+    } else {
+      params.delete('sex');
+    }
+
+    setSearchParams(params);
+  }
+
+  const handleQueryChange = (event: React.ChangeEvent<HTMLInputElement>) => {
+    const params = new URLSearchParams(searchParams);
+
+    params.set('query', event.target.value);
+    setSearchParams(params);
+  };
+
+  const handleCenturyFiltering = (
+    event: React.MouseEvent<HTMLAnchorElement>,
+    c: string,
+  ) => {
+    event.preventDefault();
+
+    const params = new URLSearchParams(searchParams);
+    const newCentury = centuries.includes(c)
+      ? centuries.filter(century => century !== c)
+      : [...centuries, c];
+
+    params.delete('century');
+    newCentury.forEach(century => params.append('century', century));
+
+    setSearchParams(params);
+  };
+
+  const handleSelectAllCentury = (
+    event: React.MouseEvent<HTMLAnchorElement>,
+  ) => {
+    event.preventDefault();
+
+    const params = new URLSearchParams(searchParams);
+
+    params.delete('century');
+    allCenturies.forEach(century =>
+      params.append('century', century.toString()),
+    );
+
+    setSearchParams(params);
+  };
+
+  const resetAll = (event: React.MouseEvent<HTMLAnchorElement>) => {
+    event.preventDefault();
+
+    const params = new URLSearchParams(searchParams);
+
+    params.delete('sex');
+    params.delete('query');
+    params.delete('century');
+
+    setSearchParams(params);
+  };
+
   return (
     <nav className="panel">
       <p className="panel-heading">Filters</p>
 
       <p className="panel-tabs" data-cy="SexFilter">
-        <a className="is-active" href="#/people">
+        <a
+          className="is-active"
+          href="#/people"
+          onClick={event => handleSexFiltering(event, null)}
+        >
           All
         </a>
-        <a className="" href="#/people?sex=m">
+        <a
+          className=""
+          href="#/people?sex=m"
+          onClick={e => handleSexFiltering(e, SexEnum.Male)}
+        >
           Male
         </a>
-        <a className="" href="#/people?sex=f">
+        <a
+          className=""
+          href="#/people?sex=f"
+          onClick={e => handleSexFiltering(e, SexEnum.Female)}
+        >
           Female
         </a>
       </p>
@@ -22,6 +111,8 @@ export const PeopleFilters = () => {
             type="search"
             className="input"
             placeholder="Search"
+            value={query}
+            onChange={handleQueryChange}
           />
 
           <span className="icon is-left">
@@ -33,52 +124,27 @@ export const PeopleFilters = () => {
       <div className="panel-block">
         <div className="level is-flex-grow-1 is-mobile" data-cy="CenturyFilter">
           <div className="level-left">
-            <a
-              data-cy="century"
-              className="button mr-1"
-              href="#/people?centuries=16"
-            >
-              16
-            </a>
-
-            <a
-              data-cy="century"
-              className="button mr-1 is-info"
-              href="#/people?centuries=17"
-            >
-              17
-            </a>
-
-            <a
-              data-cy="century"
-              className="button mr-1 is-info"
-              href="#/people?centuries=18"
-            >
-              18
-            </a>
-
-            <a
-              data-cy="century"
-              className="button mr-1 is-info"
-              href="#/people?centuries=19"
-            >
-              19
-            </a>
-
-            <a
-              data-cy="century"
-              className="button mr-1"
-              href="#/people?centuries=20"
-            >
-              20
-            </a>
+            {allCenturies.map(century => (
+              <a
+                key={century}
+                data-cy="century"
+                className={cn('button mr-1', {
+                  'is-info': centuries.includes(century.toString()),
+                })}
+                href={`#/people?${century}`}
+                onClick={e => handleCenturyFiltering(e, century.toString())}
+              >
+                {century}
+              </a>
+            ))}
           </div>
 
           <div className="level-right ml-4">
             <a
               data-cy="centuryALL"
               className="button is-success is-outlined"
               href="#/people"
+              onClick={handleSelectAllCentury}
             >
               All
             </a>
@@ -87,7 +153,11 @@ export const PeopleFilters = () => {
       </div>
 
       <div className="panel-block">
-        <a className="button is-link is-outlined is-fullwidth" href="#/people">
+        <a
+          className="button is-link is-outlined is-fullwidth"
+          href="#/people"
+          onClick={resetAll}
+        >
           Reset all filters
         </a>
       </div>