Which are the best design and user experience firms that can redesign the look, feel and information flow of a website?
Given the paucity of information here, for instance your audience (and industry), the site goals, your budget, the metrics of your current site, and the talent that you have in house as well as the last time you did a redesign, I am going to assume a number of things, as well as time and budget being infinite, the site being mostly content/marketing with little transactions, your team being small that can be dedicated full time to this project. For the top website UX/Visual Communication Design firms, you must expect to pay no less than $100K. You must also expect (assuming the taxonomy and content strategy of your current site is not massive), that it should cost no more than $250K. Mind you - this is for the very best - this is not Ford Focus territory, this is Astin Martin.
Two notes (These are strong opinions, so take that for what it's worth - but it's also based on 15 years experience working for product companies, operating companies, start-ups, enterprises, as well as for a few "digital" agencies):
First, do not, under any circumstances, go to a top tier "digital" agency. Your budget will be blown on account directors, fancy offices, lush travel accounts, and the direction and design will be led by the vicissitudes of a cranky creative director who has more interest in pursuing a dream of producing the next great independent firm than designing an elegant, useful, usable, desirable website that meets your core business objectives. Large digital agencies, by and large, also don't have a core UX capability - their dirty little secret is they outsource most of the usability, information architecture, interaction design, and user research to contractors. A simple test is to look at the senior leadership of the firm - if there is an executive creative director, but no senior UX person on the leadership team, you know what their focus is - internal power structure indicates values - and if UX has no power in the organization, it's an after-thought, and it will be an after-thought on the design of your site.
Second, do not hire a development firm that advertises their expertise in "Rails, Python, iOS, ,insert language and/or platform here>", because while they may be able to write elegant, standards compliant code in a rapid manner, they will not, by and large, have a focus on user research, audience definition, design testing, or visual communication design. If the firm is dedicated to Adobe Flash implementations, run - don't walk.
The top user experience design and visual communication design firms (notice the difference), for that kind of project with 3-6 months, and over 100K in budget is hands-down Happy Cog, Messagefirst, and Behavior Design. This is my opinion, but each of these firms has a strong reputation in both the design and user experience communities, have delivered consistent results year-over-year, and have processes and methodologies to back up their work as well as in-house talent in leadership roles that are considered best of the best by their respective communities. None of them will break your bank either (you can definitely spend more).
http://happycog.com/
http://messagefirst.com/
http://behaviordesign.com
Now, if you are loaded, and want to spend either that same amount, or even significantly more for the exact same quality or slightly lower quality site, then by all means, go with one of these next three, that will still deliver relatively good results, but more of your budget will end up funding fancy offices and multiple account directors (with all they lovely padding that comes with it), but not necessarily any meaningful or measurable increase in the value of your new site:
Huge, Inc
R/GA
AKQA
Each of these firms will design something that is good, but ceteris paribus, are at least 2x more expensive than the previous 3 firms, with no significant increase in the value of the product they will deliver to you. They do have quality UX and Creative in-house (will not out-source it to contractors, and they have in-house UX and Creative talent that are well-known and respected in their communities - perhaps not to the extent of the first three - but solid), but their rate sheets may bring tears to your eyes.
Happy hunting. If you want the same breakdown for transaction-heavy sites, e-commerce sites, or massive content sites, I can provide that information as well.
One last note: I know sometimes Forrester Research publishes something related to Web Design or User Experience Design, and in the course of doing an industry overview of the fields and disciplines, sometimes goes as far as recommending a few firms that do ux and/or web design work. I would take their recommendations with many, many grains of salt. Firstly, they are not known for doing solid, quantifiable research in the UX field; also, they are not exactly considered the arbiters of great ux, design principles, etc - and finally, many times they have been known for recommending firms or agencies that just happen to be clients of theirs - so they are not the most objective source of information or recommendations - even if they were considered thought leaders in the ux space - which they are not. I once saw a "heuristic evaluation" done by them which was, in fact, done by people for the ad agency world - and let's just say that their understanding of design principles, user-centered design, and best practices was bordering on negligent idiocy. #JustSaying
Again, just my opinion, YMMV.
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